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

72 CHAPTER 5: Estimating of the Jet Energy Scale Calibration FactorFigure 5.3: A simple example to show the basic concepts of the MVA method. In thisexample, the circles, which are representing events, are observed in two different bins,being either “1” or “2” and can belong to one of “S” or “B” classes.p(S) = 5 10 ,p(B) = 510 .From the above elements, p(1) and p(2) can also be extracted, according to the “sumrule” of the probability theory, as followsp(1) = p(1|S)p(S) + p(1|B)p(B) = 2 5 ,p(2) = p(2|S)p(S) + p(2|B)p(B) = 3 5 .Now that all the input information is complete, one can calculate the posterior probabilitiesas expressed belowp(S|1) = p(1|S)p(S)p(1)= 912 ,p(S|2) = p(2|S)p(S)p(2)p(B|1) = p(1|B)p(B)p(1)p(B|2) = p(2|B)p(B)p(2)= 412 ,= 312 ,= 812 .The above numbers can be interpreted as follows. Any new event, whose measured Xvalue yields to X = 1, would be assigned to class S since the probability of being atype “S” is three times more than being a type “B” event when a value equals 1 ismeasured. With the same reasoning, any new event is grouped to the class B giventhat the meaurement of the X property of that event results in X = 2.The above example shows the basic idea of assigning a new observed event to a specificclass in a one-dimensional phase space. In most cases, usually more than one inputvariable is used. As a result, the problem of labeling an event as signal or background,would not be that simple. Defining a single variable out of many input variables canprovide a possible solution. Hence the Likelihood function L(⃗x), which combines the