ROOT and statistics tutorial Exercise: Discover the Higgs, part 2

1. ROOT and statistics tutorialExercise: Discover the Higgs, part 2 Attilio Andreazza Università di Milano and INFN CaterinaDoglioni Université de Genève

2. Outline • Whatwillwe do today: • Discover the Higgsboson of course! ...well, check ATLAS isnotcheatingus • Whatwewilllearn! • Computing confidencelevelsbased on Poissonianstatistics. • Compute confidencelevelsbasedif the expectedm(B,S)isuncertain. • For peoplerunning fast: • Expected 95% levelexclusion + errorbars 2011+2012 data In the region 125 ± 5 GeV Dataset 2011 2012 2011+2012 Expected B only 2±0.3 3±0.4 5.1±0.8 Expected S mH=125 GeV2±0.3 3±0.5 5.3±0.8 Observed in the data 4 9 13 From F. Gianotti’sHiggs seminar Root and statistics tutorial: Exercises

3. Confidenceleveldefinition • Definition of confidencelevel • N.B.:thisisthe frequentistdefinition, not the Bayesianone from the lectures, butthatallowsyou to makeallcomputations by yourself and to grasp the mainfeatures of the problem. • A certaincriterionrejects an hypothesis with C.L. aif, in case the hypothesisistrue, itwould be erroneouslyrejected by thatcriterion on a fraction 1-a of the cases. • In ourexercise: • Weobserve a certainnumber of eventsNobs • Weexpecta certainnumber of background events:NB • Discovery: wereject the hypothesisour sample containsonlybackground eventsif: • Exclusion:wereject an hypothesisexpectingNSsignaleventsif: Root and statistics tutorial: Exercises

4. Confidencelevelcomputation • At first just assume a Poissonstatistics: • We can compute the probabilitysumming the probabilities of allN up to Nobs • Butwewant to use a Monte Carlo method!Why? Itwill be easier to extend to the treatment of systematicuncertainties (thisiswhat BAT or RooStats do for example). • What to do: • Sample the probabilitydistribution M times (say M=10000) • Counthowmanycases M’ the value of Nexceeds (or islowerthan, ifappopriated) Nobs. • We can reject the hypothesisif M’/M<1-a. • Somethinglike: Int_t M=0; for (Int_t i=0; i<10000; i++) { Int_tN = gen.Poisson(NB); if ( N>=Nobs ) M++; } Double_t CL = 1.-M/10000. Root and statistics tutorial: Exercises

5. Computing confidencelevels • Neglectinguncertainties on NB • With which CL can wereject the background-onlyhypothesiswhenusing: • Only 2011 data • Only 2012 data • The combined set • Ifonewouldhavebeenobservedonly the expectedbackgroun (i.e. Nobs =2,3 and 5 eventsrespectively for 2011, 2012 and combineddataset), with whichconfidencelevelonewouldhaverejected the hypothesis of the Higgspresence? • Repeatadding the uncertainty on the mvalue of the Poissondistribution. Assume the uncertainties on NB and NS are fullycorrelated. • In such a situation and the integral can be performed by samplingNB from a Gaussiandistribution and afterwardsamplingN. Root and statistics tutorial: Exercises

6. Computing exclusionlimit • Observedexclusionlimit: • aftergettingNobsevents, allNS>N0,S are excludedatconfidencelevela, whereis the N0,S minimum onesatisfying the relation: • Determiningthis minimum, even in thissimple case isquitecomputationallyexpensive, and special tools are usuallyemployed. • Expectedexclusionlimit: • Is the onethatwouldobtainedifNobswouldcorrespond to the median of • The ±1sexpectedvaluescorrespond to the limitthatwould be obtainedifNobswouldcoincide with the 16% and 84% percentiles of • The ±1sexpectedvaluescorrespond to the limitthatwould be obtainedifNobswould coincide with the 2.25% and 97.75% percentiles of • Compute the expectedlimits for ATLAS and mH=125 GeV Root and statistics tutorial: Exercises

7. 2011 data 2012 data • Repeating the samecalculation for allmHhypothesisis the way the famous band plots are produced. • From F. Gianotti’sHiggs seminar Excluded (95% CL): 131-162, 170-460 GeV Expected: 124-164, 176-500 GeV Excluded (95% CL): 130-170 GeV Expected: xxxxGeV 2011+2012 data