rpp2010-rev-statistics - 33 Statistics 1 33 STATISTICS...

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33. Statistics 1 33. STATISTICS Revised September 2009 by G. Cowan (RHUL). This chapter gives an overview of statistical methods used in high-energy physics. In statistics, we are interested in using a given sample of data to make inferences about a probabilistic model, e.g. , to assess the model’s validity or to determine the values of its parameters. There are two main approaches to statistical inference, which we may call frequentist and Bayesian. In frequentist statistics, probability is interpreted as the frequency of the outcome of a repeatable experiment. The most important tools in this framework are parameter estimation, covered in Section 33.1, and statistical tests, discussed in Section 33.2. Frequentist confidence intervals, which are constructed so as to cover the true value of a parameter with a specified probability, are treated in Section 33.3.2. Note that in frequentist statistics one does not define a probability for a hypothesis or for a parameter. Frequentist statistics provides the usual tools for reporting the outcome of an experiment objectively, without needing to incorporate prior beliefs concerning the parameter being measured or the theory being tested. As such, they are used for reporting most measurements and their statistical uncertainties in high-energy physics. In Bayesian statistics, the interpretation of probability is more general and includes degree of belief (called subjective probability). One can then speak of a probability density function (p.d.f.) for a parameter, which expresses one’s state of knowledge about where its true value lies. Bayesian methods allow for a natural way to input additional information, such as physical boundaries and subjective information; in fact they require the prior p.d.f. as input for the parameters, i.e. , the degree of belief about the parameters’ values before carrying out the measurement. Using Bayes’ theorem Eq. (32 . 4), the prior degree of belief is updated by the data from the experiment. Bayesian methods for interval estimation are discussed in Sections 33.3.1 and 33.3.2.6 Bayesian techniques are often used to treat systematic uncertainties, where the author’s beliefs about, say, the accuracy of the measuring device may enter. Bayesian statistics also provides a useful framework for discussing the validity of different theoretical interpretations of the data. This aspect of a measurement, however, will usually be treated separately from the reporting of the result. For many inference problems, the frequentist and Bayesian approaches give similar numerical answers, even though they are based on fundamentally different interpretations of probability. For small data samples, however, and for measurements of a parameter near a physical boundary, the different approaches may yield different results, so we are forced to make a choice. For a discussion of Bayesian vs. non-Bayesian methods, see References written by a statistician[1], by a physicist[2], or the more detailed comparison in Ref. [3].
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