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Program for evaluation of the significance, confidence intervals and limits by direct probabilities calculations S.Bityukov (IHEP,Protvino) , S.Erofeeva(MSA IECS,Moscow), N.Krasnikov(INR RAS, Moscow) , A.Nikitenko(IC, London)

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Program for assessment of the hugeness, certainty interims and points of confinement by direct probabilities figurings S.Bityukov (IHEP,Protvino) , S.Erofeeva(MSA IECS,Moscow), N.Krasnikov(INR RAS, Moscow) , A.Nikitenko(IC, London) September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Introduction During arranging or handling of investigation we regularly consider a measurable theory H 0 : new material science is available in Nature against speculation H 1 : new material science is missing in Nature . The estimation of vulnerability in our decision is characterized by the probabilities ï¡ = P(reject H 0 | H 0 is genuine) - Type I lapse and b = P(accept H 0 | H 0 is false) - Type II blunder There are numerous meanings of essentialness as a measure of abundance of sign occasions above foundation. Numerous methodologies exist likewise to routines for development of interims and points of confinement: certainty, tolerant, fiducial etc. Amid one of the CMS gatherings Gunter Quast planned the issue of practicians âthe just remaining issue: settle on a decision â¦ picked system ought to be â as basic as could be expected under the circumstances, however not wrong !â September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Motivation of the work (criticalness) Gaussian point of confinement gives the wrong response for low estimation of Î² (tail of Poisson appropriation is heavier than tail of Gaussian ) 2. The measurements like S L (a probability proportion based test measurement) have poor factual properties as estimator of centrality ( S L = â2â¢(ln L1-ln L2)= â2â¢(ln Q), where Q is the proportion of binned/unbinned probability fits for speculations H 0 and H 1 H 0 : sign present and H 1 : no sign present) The least difficult noteworthiness is the criticalness S_cP portrayed at the following slide. The S_cP is very normal centrality , which permits to consider any vulnerabilities by direct figuring of probabilities. September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Definition of the criticalness S_cP - likelihood from Poisson appropriation with mean Î¼_ b to watch equivalent or more prominent than Nobs occasions, changed over to proportionate number of sigmas of a Gaussian dispersion (see report (page 8) by G. Quast in CMS Physics examination days, May 9-12, 2005, CERN http://cmsdoc.cern.ch/~bityukov/talks/talks.html likewise, see I.Narsky, NIM A450(2000)444). The displayed project ScP permits to ascertain this essentialness with considering exploratory systematics with factual properties ( Gaussian rough guess) and hypothetical systematics with no measurable properties. Likewise, the system figures (if alternative is on) the joining hugeness of a few channels. As is expected all channels are free. September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Conception. The likelihood of making a Type II lapse ( Î² ) in speculations test about vicinity of sign in analysis ( H0 ) is utilized for determination number of sigmas (of foundation conveyance) between expected foundation and watched number of occasions Nobs (recipe 8 in CMS CR 2002/05 ). This likelihood is utilized for determination of sign criticalness, i.e. the essentialness S_cP will be found under determining of mathematical statements , where It can be utilized as a part of consolidating of results. Give us a chance to consider two conceivable methodologies. September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Combining of watched results: Approach 1 Approach 1. Assume that watched quality is more noteworthy than anticipated foundation Let Î² _1 be Type II blunder for channel 1 (occasion A = foundation ï³ Nobs _1 with P( A )= Î² _1 ) and Î² _2 be Type II mistake for channel 2 (occasion B = foundation ï³ Nobs _2 with P( B )= Î² _2 ). Since occasion An is free from occasion B then likelihood of synchronous appearance of An and B meets Î² _12 = P( AB ) = P( A )*P( B ) = Î² _1* Î² _2 . After determination of Î² _12 one can compute the S_cP . September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Combining of expected signs & foundations: Approach 2 Approach 2. Assume that Nobs is normal aggregate of expected sign ( Î¼_ s ) and expected foundation ( Î¼_b ), i.e. Nobs = Î¼_ s+ Î¼_b (the instance of arranged trial). At that point the wholes of expected quantities of sign ( Î¼_s_i ) and foundation ( Î¼_b_i ) occasions in every channel are utilized as rundown Î¼_s and Î¼_b for count of consolidated criticalness. Note that we consider for this situation as change of expected foundation and vacillation of expected sign. After determination of relating Î² one can ascertain the S_cP . September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Uncertainties The project considers two sorts of instabilities: exploratory systematics with measurable properties (we accept that this systematics has Gaussian dispersion with known fluctuation Ï _b**2 in concurring with recipe Î¼_ b = expected foundation + N (0, Ï _b) ). Appr.2: For the joining's situation of channels the outline difference Ï _b**2 is the total of fractional change Ï _b_i**2 . b) hypothetical systematics ( Î' _b ) with no factual properties (we expect: the most pessimistic scenario happens when the foundation is maximal, i.e. Î¼_ b*(1+ Î' _b) , however we take the sign in addition to the foundation as Nobs ; more data can be found in S.Bityukov, N.Krasnikov, CMS CR 2002/05 or S.Bityukov, N.Krasnikov, Mod.Phys.Lett.A 13 (1998)3235) Appr.2: The consolidating Î' _b is the entirety of halfway Î' _b_i . September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Main data and yield parameters Main information parameters: 1. expected foundation â Î¼_ b 2. signal = watched worth ( Nobs ) - expected foundation ( Î¼_ b ) â Î¼_ s 3. exploratory vulnerability ( r.m.s.) of foundation with measurable properties â Ï_ b 4. systematics of hypothetical root in foundation â Î'_ b Output parameters: 1. importance S_cP , ascertained by equation - dsgnf 2. noteworthiness S_cP_MC , computed by Monte Carlo - dsgnfm September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Auxiliary info parameters 1. switch for picking of sort counts - iflag = 1 figurings by recipe (snappy computations) iflag = 2 Monte Carlo estimations iflag = 12 estimations by equation and by Monte Carlo 2. number of channels for figurings - nchan (from 1 up to 10 ) 3. number of channels for joined S_cP - ncombi (from 1 up to nchan ) 4. parameter for Monte Carlo estimation - over - parameter for Monte Carlo figurings. It is various Monte Carlo trials which will give estimation of number occasions over or break even with Nobs . This parameter (and inward esteem dbeta ) decides the quantity of trials for given Î¼_ s, Î¼_ b, Ï _b and Î'_ b in routine SCPMC . September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

The structure of system Language: Fortran 77 iflag Three distinct sorts 1. SCPFOR - figurings by recipe of computations: 2. SCPMC - Monte Carlo counts 12. SCPFOR + SCPMC Main project forms the client prerequisites (characterized in administrators DATA) and calls schedules SCPFOR and/or SCPMC . September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Problem and rough guess The issue which happens amid computations is the limited scope of materialness of standard method DGAUSN in CERNLIB . For estimations of S_cP>6.2-7 the strategy gives non right result. For this situation we use as a decent close estimation the hugeness (MPL A13 (1998)3235) S_c12 = 2 ( ï( Î¼_ s+ Î¼_b) - ï Î¼_ b) . The instabilities' record is extremely straightforward: hypothetical systematics ( Î' _b ) S_c12t = 2 ( ï( Î¼_ s+ Î¼_b) - ï( Î¼_ b(1+ Î' _b)) . test systematics ( Ï _b**2 ) ï Î¼_ b S_c12e = 2 ( ï( Î¼_ s+ Î¼_b) - ï Î¼_ b) - - . ï( Î¼_ b+ Ï _b**2) September, 2005 PhyStat 2005 Oxford, UK S.Bityukov

Simplest sample of project S_cP yield Example of G.Quast: bkg= 2 sig= 5.4 . Here S 1 =3.8 S 12 =2.6 S L =2.7 Significance S_cP and/or S_cP_MC: NN of channels = 1 , Combining channels