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1 Statistical Problems in Particle Physics Louis Lyons Oxford IPAM, November 2004

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2 HOW WE MAKE PROGRESS Read Statistics books Kendal + Stuart Papers, internal notes Feldman-Cousins, Orear,…….. Experiment Statistics Committees BaBar, CDF Books by Particle Physicists Eadie, Brandt, Frodeson, Lyons, Barlow, Cowan, Roe,… PHYSTAT series of Conferences

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3 History of Conferences Overview of PHYSTAT 2003 Specific Items Bayes and Frequentism Goodness of Fit Systematics Signal Significance At the pit-face Where are we now ? PHYSTAT

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4 WhereCERNFermilabDurhamSLAC When Jan 2000March 2000March 2002Sept 2003 Issues Limits Wider range of topics Physicists Particles +3 astrophysicists Particles +3 astrophysicists Particles + Astro + Cosmo Statisticians 3 3 2Many HISTORY

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5 Future PHYSTAT05 Oxford, Sept 12 th – 15 th 2005 Information from Limited to 120 participants Committee: Peter Clifford, David Cox, Jerry Friedman Eric Feigelson, Pedro Ferriera, Tom Loredo, Jeff Scargle, Joe Silk

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6 Issues Bayes versus Frequentism Limits, Significance, Future Experiments Blind Analyses Likelihood and Goodness of Fit Multivariate Analysis Unfolding At the pit-face Systematics and Frequentism

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7 Talks at PHYSTAT Introductory Talks 8 Invited talks by Statisticians 8 Invited talks by Physicists 47 Contributed talks Panel Discussion Underlying much of the discussion: Bayes and Frequentism

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8 Invited Talks by Statisticians Brad EfronBayesian, Frequentists & Physicists Persi DiaconisBayes Jerry FriedmanMachine Learning Chris GenoveseMultiple Tests Nancy ReidLikelihood and Nuisance Parameters Philip StarkInference with physical constraints David VanDykMarkov chain Monte Carlo John RiceConference Summary

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9 Invited Talks by Physicists Eric FeigelsonStatistical issues for Astroparticles Roger BarlowStatistical issues in Particle Physics Frank PorterBaBar Seth DigelGLAST Ben WandeltWMAP Bob NicholData mining Fred JamesTeaching Frequentism and Bayes Pekka SinervoSystematic Errors Harrison ProsperMultivariate Analysis Daniel StumpPartons

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10 Bayes versus Frequentism Old controversy Bayes 1763 Frequentism 1937 Both analyse data (x) statement about parameters ( ) e.g. Prob ( ) = 90% but very different interpretation Both use Prob (x; )

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11 Bayesian posteriorlikelihoodprior Problems: P(param) True or False “Degree of belief” Prior What functional form? Flat? Which variable? Unimportant when “data overshadows prior” Important for limits Bayes Theorem

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12 P (Data;Theory) P (Theory;Data) HIGGS SEARCH at CERN Is data consistent with Standard Model? or with Standard Model + Higgs? End of Sept 2000 Data not very consistent with S.M. Prob (Data ; S.M.) < 1% valid frequentist statement Turned by the press into: Prob (S.M. ; Data) < 1% and therefore Prob (Higgs ; Data) > 99% i.e. “It is almost certain that the Higgs has been seen”

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13 P (Data;Theory) P (Theory;Data) Theory = male or female Data = pregnant or not pregnant P (pregnant ; female) ~ 3% but P (female ; pregnant) >>>3%

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14 at 90% confidence and known, but random unknown, but fixed Probability statement about and Frequentist Bayesian and known, and fixed unknown, and random Probability/credible statement about

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15 Bayesian versus Frequentism Basis of method Bayes Theorem --> Posterior probability distribution Uses pdf for data, for fixed parameters Meaning of probability Degree of beliefFrequentist defintion Prob of parameters? YesAnathema Needs prior?YesNo Choice of interval? YesYes (except F+C) Data considered Only data you have….+ more extreme Likelihood principle? YesNo Bayesian Frequentist

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16 Bayesian versus Frequentism Ensemble of experiment NoYes (but often not explicit) Final statement Posterior probability distribution Parameter values Data is likely Unphysical/ empty ranges Excluded by priorCan occur SystematicsIntegrate over priorExtend dimensionality of frequentist construction CoverageUnimportantBuilt-in Decision making Yes (uses cost function)Not useful Bayesian Frequentist

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17 Bayesianism versus Frequentism “Bayesians address the question everyone is interested in, by using assumptions no-one believes” “Frequentists use impeccable logic to deal with an issue of no interest to anyone”

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18 Goodness of Fit Basic problem: very general applicability, but Requires binning, with > 5…..20 events per bin. Prohibitive with sparse data in several dimensions. Not sensitive to signs of deviations K-S and related tests overcome these, but work in 1-D So, need something else.

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19 Goodness of Fit Talks Zech Energy test Heinrich Yabsley & Kinoshita ? Raja Narsky What do we really know? Pia Software Toolkit for Data Analysis Ribon ……………….. Blobel Comments on minimisation

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20 Goodness of Fit Gunter Zech“Multivariate 2-sample test based on logarithmic distance function” See also: Aslan & Zech, Durham Conf., “Comparison of different goodness of fit tests” R.B. D’Agostino & M.A. Stephens, “Goodness of fit techniques”, Dekker (1986)

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21 Likelihood & Goodness of Fit Joel Heinrich CDF note #5639 Faulty Logic: Parameters determined by maximising L So larger is better So larger implies better fit of data to hypothesis Monte Carlo dist of for ensemble of expts

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22 not very useful e.g. Lifetime dist Fit for i.e. function only of t Therefore any data with the same t same so not useful for testing distribution (Distribution of due simply to different t in samples)

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23 SYSTEMATICS For example Observed for statistical errors Physics parameter we need to know these, probably from other measurements (and/or theory) Uncertainties error in Some are arguably statistical errors Shift Central Value Bayesian Frequentist Mixed

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24 Simplest Method Evaluate using and Move nuisance parameters (one at a time) by their errors If nuisance parameters are uncorrelated, combine these contributions in quadrature total systematic Shift Nuisance Parameters

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25 Bayesian Without systematics prior With systematics Then integrate over LA and b

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26 If = constant and = truncated Gaussian TROUBLE! Upper limit on from Significance from likelihood ratio for and

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27 Frequentist Full Method Imagine just 2 parameters and LA and 2 measurements N and M PhysicsNuisance Do Neyman construction in 4-D Use observed N and M, to give Confidence Region for LA and LA 68%

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28 Then project onto axis This results in OVERCOVERAGE Aim to get better shaped region, by suitable choice of ordering rule Example: Profile likelihood ordering

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29 Full frequentist method hard to apply in several dimensions Used in 3 parameters For example: Neutrino oscillations (CHOOZ) Normalisation of data Use approximate frequentist methods that reduce dimensions to just physics parameters e.g. Profile pdf i.e. Contrast Bayes marginalisation Distinguish “profile ordering” Properties being studied by Giovanni Punzi

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30 Talks at FNAL CONFIDENCE LIMITS WORKSHOP (March 2000) by: Gary Feldman Wolfgang Rolk p-ph/ version 2 Acceptance uncertainty worse than Background uncertainty Limit of C.L. as Need to check Coverage

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31 Method: Mixed Frequentist - Bayesian Bayesian for nuisance parameters and Frequentist to extract range Philosophical/aesthetic problems? Highland and Cousins NIM A320 (1992) 331 (Motivation was paradoxical behaviour of Poisson limit when LA not known exactly)

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32 Systematics & Nuisance Parameters SinervoInvited Talk (cf Barlow at Durham) BarlowAsymmetric Errors Dubois-FelsmannTheoretical errors, for BaBar CKM CranmerNuisance Param in Hypothesis Testing Higgs search at LHC with uncertain bgd RolkeProfile method see also: talk at FNAL Workshop and Feldman at FNAL (N.B. Acceptance uncertainty worse than bgd uncertainty) DemortierBerger and Boos method

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33 Systematics: Tests Do test (e.g. does result depend on day of week?) Barlow: Are you (a) estimating effect, or (b) just checking? If (a), correct and add error If (b), ignore if OK, worry if not OK BUT: 1)Quantify OK 2)What if still not OK after worrying? My solution: Contribution to systematics’ variance is even if negative!

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34 Barlow: Asymmetric Errors e.g. Either statistical or systematic How to combine errors ( Combine upper errors in quadrature is clearly wrong) How to calculate How to combine results

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35 Significance Significance = ? Potential Problems: Uncertainty in B Non-Gaussian behavior of Poisson Number of bins in histogram, no. of other histograms [FDR] Choice of cuts (Blind analyses Choice of bins Roodman and Knuteson) For future experiments Optimising could give S =0.1, B = 10 -6

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36 Talks on Significance GenoveseMultiple Tests LinnemannComparing Measures of Significance RolkeHow to claim a discovery ShawhanDetecting a weak signal TerranovaScan statistics QuayleHiggs at LHC PunziSensitivity of future searches BityukovFuture exclusion/discovery limits

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37 Multivariate Analysis FriedmanMachine learning ProsperExperimental review CranmerA statistical view LoudinComparing multi-dimensional distributions RoeReducing the number of variables (Cf. Towers at Durham) HillOptimising limits via Bayes posterior ratio Etc.

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38 From the Pit-face Roger BarlowAsymmetric errors William Quayle Higgs search at LHC etc. From Durham: Chris ParkesCombining W masses and TGCs Bruce Yabsley Belle measurements

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39 Blind Analyses Potential problem: Experimenters’ bias Original suggestion? Luis Alvarez concerning Fairbank’s ‘discovery’ of quarks Aaron Roodman’s talk Methods of blinding: Keep signal region box closed Add random numbers to data Keep Monte Carlo parameters blind Use part of data to define procedure Don’t modify result after unblinding, unless………. Select between different analyses in pre-defined way See also Bruce Knuteson: QUAERO, SLEUTH, Optimal binning

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40 Where are we? Things that we learn from ourselves –Having to present our statistical analyses Learn from each other –Likelihood not pdf for parameter Don’t integrate L –Conf int not Prob(true value in interval; data) –Bayes’ theorem needs prior –Flat prior in m or in are different –Max prob density is metric dependent –Prob (Data;Theory) not same as Prob(Theory;Data) –Difference of Frequentist and Bayes (and other) intervals wrt Coverage –Max Like not usually suitable for Goodness of Fit

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41 Where are we? Learn from Statisticians –Update of Current Statistical Techniques –Bayes: Sensitivity to prior –Multivariate analysis –Neural nets –Kernel methods –Support vector machines –Boosting decision trees –Hypothesis Testing : False discovery rate –Goodness of Fit : Friedman at Panel Discussion –Nuisance Parameters : Several suggestions

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42 Conclusions Very useful physicists/statisticians interaction e.g. Upper Limit on Poisson parameter when: observe n events background, acceptance have some uncertainty For programs, transparencies, papers, etc. see: Workshops: Software, Goodness of Fit, Multivariate methods,… Mini-Workshop: Variety of local issues Future: PHYSTAT05 in Oxford, Sept 12 th – 15 th, 2005 Suggestions to:

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