Measurement of the Jet Energy Scale in the CMS experiment ... - IIHE

CHAPTER 5: Estimating of the Jet Energy Scale Calibration Factor 71The ratio y L is then calculated for every jet combination in each event. Per event, thejet combination whose y L is the largest, is chosen and returned by the MVA method.Almost all the MVA methods which includd a Likelihood Ratio method need to betrained before applying the method on data. In the training phase, one introducesthe signal and background PDFs to the MVA method. Then the MVA method trainsitself and learns what kind of behaviours can be extracted from both the signal andbackgrounds PDFs. In the application phase, using the information which has beencollected in the training phase, the MVA method returns a value per event and categorizesthat event as signal or background. In case of a jet combination study, where onesignal against eleven backgrounds is present per event, the signal is defined as the jetcombination with the highest value returned by the MVA method, although in somecases the MVA method is not able to return the “true” jet combination, which correspondsto the correct jet combination that is matched with the hard-scatter partons.5.1.3 Likelihood Concept in Bayesian StatisticsAccording to the Bayes’ theorem, posterior probability p(Y |X), is related to priorprobability p(Y ), with the following equationp(Y |X) =p(X|Y )p(Y ),p(X)where p(X|Y ) is the conditional probability and is also referred to as the Likelihood. Inthe above equation, p(X|Y ) is the probability distribution of the parameter X, whichis usually a continuous variable of the event that belongs to the class Y and p(Y |X) isthen defined as the probability of assigning a new observed event to the class Y giventhat the value X is measured for that particular event. Therefore in order to obtain theposterior probability p(Y |X), in addition to the prior probability p(Y ), p(X|Y ) shouldalso be known. In the language of multi-variate techniques, p(X|Y ) is obtained in thetraining phase. During training, the classes Y with their properties X are introducedto the trainer and subsequently the corresponding PDFs are extracted.In order to clarify a bit more how a generic multi variate method works, a simplifiedexample based on Bayes’ theorem is explained here. Consider a space with one variableX that can take only two values, namely “X = 1” or “X = 2”. In this space, eventsare categorized in either signal or background classes, hence “Y = S” or “Y = B”. Anevent is said to be measured when the X value of that particular event is determined.Assume 10 such events are selected and fed to an MVA method. The results of trainingover 10 events, are summarized in Figure 5.3.The information, containing the conditional as well as the prior probabilities, thatcan be derived from Figure 5.3, is listed below.p(1|S) = 3 5 , p(1|B) = 1 5 ,p(2|S) = 2 5 , p(2|B) = 4 5 ,