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Statistics Forum News

This forum news article from December 2008 discusses the statistical methods used for combining search channels in the Higgs boson discovery. It applies these methods to specific Higgs channels and treats systematics using the profile likelihood method. The article also mentions ongoing developments in Bayesian methods and the look-elsewhere-effect. The note is part of the CSC book and provides valuable insights for researchers in the field.

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Statistics Forum News

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  1. Statistics Forum News Glen Cowan Eilam Gross ATLAS Statistics Forum CERN, 3 December, 2008

  2. Recent Statistics Forum activity Higgs Combination Exercise  CSC note HG11 completed Statistics Book  Statistics FAQ Interactions with physics groups ATLAS/CMS Joint Statistics Meetings

  3. Higgs combination complete (p 1473 of CSC book) Describes statistical methods for combination of search channels. Applies combination to (a subset of) Higgs channels: WW→e(0-jet and 2-jet), , ,ZZ(*)→4l Treats systematics by means of profile likelihood method. Considers fixed-mass hypothesis; look-elsewhere-effect must be studied separately. NB H and WW CSC notes describe floating-mass fits. Some approximations used for discovery/exclusion significance. Valid for L > 2 fb. For lower L need toy MC methods. Does not represent final word on methods -- other developments ongoing (Bayesian, CLs, look-elsewhere-effect,...)

  4. CSC HG11 People Editors: Eilam Gross, Glen Cowan Contacts for individual channels: WW Bill Quayle  Yaquan Fang  Kyle Cranmer 4l Andrea Di Simone Note referee: Fabiola Gianotti Chapter referees: Fabio Cerutti, Bruno Mansoulie Chapter editors: Rachid Mazini, Bill Murray + e.g. Louis Fayard, Aleandro Nisati, Karl Jakobs, Dave Charlton, Ofer Vitells...

  5. Reminder of combination method The single-channel likelihood function uses Poisson model for events in signal and control histograms:  represents all nuisance parameters, e.g., background rate, shapes  =1 is SM.  =0 is background only. There is a likelihood Li(,i) for each channel, i = 1, …, N. The full likelihood function is

  6. Profile likelihood ratio To test hypothesized value of , construct profile likelihood ratio: Maximized L for given  Maximized L Equivalently use q =  2 ln (): data agree well with hypothesized →q small data disagree with hypothesized →q large

  7. p-value / significance of hypothesized  Test hypothesized  by giving p-value, probability to see data with ≤ compatibility with  compared to data observed: Equivalently use significance:

  8. Sampling distribution of q To compute p-value of hypothesized , need distribution f (q|). For 5discovery we need a p-value of 2.8 × 10; To estimate this using MC needs > 108 events. Wilk’s theorem: in large sample limit f (q|) should approach with w = ½. (a “half-chi-square” distribution). Validation exercise shows approximation OK for L≥ 2 fb. We use the half-chi-square approximation in the current note; for lower luminosities we will need MC to get f (q|) (feasible for exclusion limits at 95% CL).

  9. Example from validation exercise: ZZ(*)→ 4l Distributions of q0 for 2, 10 fb from MC compared to ½2 ½2 ½2 (One minus) cumulative distributions. Band gives 68% CL limits. ½2 ½2 5 level

  10. Sensitivity Discovery: Generate data under s+b (= 1) hypothesis; Test hypothesis  = 0 →p-value →Z. Exclusion: Generate data under background-only (= 0) hypothesis; Test hypothesis . If = 1 has p-value < 0.05 exclude mH at 95% CL. Estimate median significance by setting data equal to expectation values (Asimov data) or by using MC. For median, can combine significances of individual channels. For significance of e.g. a real data set, need global fit.

  11. Combined discovery sensitivity 5s

  12. Combined discovery sensitivity vs. mH, L Approximations not reliable below 2 fb-1 (usually conservative)

  13. p-value of SM hypothesis (m=1) vs. mH mH > 115 at 95% CL with 2 fb-1

  14. Combined exclusion sensitivity vs. mH, L Approximations not reliable below 2 fb-1 (usually conservative)

  15. Summary of the Higgs Combination Final draft on http://indico.cern.ch/conferenceDisplay.py?confId=25831 higgs_combo_csc_28nov08.pdf CSC book imminent, we are provisionally pages 1473 to 1516 Many issues/problems identified and worked through useful. Only subset of methods explored  much more work left. Method now documented, global fit tools under development. We need to learn how to do this more quickly...

  16. Statistics Forum Website: FAQ Some general items: PDG Chapters, Pedestrian's guide, Glossary, ...

  17. Statistics Forum FAQ Notes This is a living document

  18. Statistics Forum FAQ Notes The “FAQ” consists of a collection of notes on specific questions use cases, examples, ... Bayesian methods for ATLAS Higgs search (GC) Comparison of significance from profile and integrated likelihoods (GC, EG) Discovery significance with statistical uncertainty in the background estimate (EG, OV, GC) Error analysis for efficiency (GC) How to measure efficiency (DC) MC statistical errors in ML fits (GC) Covariance matrix for histogram made using seed events (GC) If you have a note which you think should be included here, or if you are interested to write such a note or comment on a note or request a note on a specific subject please let us know.

  19. Interaction with physics groups Over past year much interaction with Higgs group Combination exercise (relevant to other searches as well...) Continue to extend this to other groups (Exotics, SUSY, B-Physics,...) E.g. SUSY Group (Dan T.) posed several question which we have tried to answer in notes (see FAQ – Profile vs integrated likelihoods, correlations in histograms arising from resampling). Encourage talks from physics groups in Statistics Forum, e.g., Exotics Lepton+X (Luis F., 24.9.08) – discussions ongoing SUSY (Sascha C., today) B-Physics (Emmerich K., 28.7.08) – ongoing Will push for further interaction – written note is preferred outcome.

  20. Joint ATLAS/CMS Statistics Forum Meetings: 25 September 2008 29 July 2008 4 September 2007 17 July 2007 Summer 08: agree to develop RooStats as common framework. Keep eye on ability to carry out independent validation. Key players: Kyle Cranmer (ATLAS) Gregory Schott (CMS) Wouter Verkerke (RooFit) Lorenzo Moneta (Root) Ongoing discussions on methodology Ideal is to use several methods (profile likelihood, Bayesian, Cls,...) for each result.

  21. Summary Higgs combination exercise complete Valuable exercise especially for developing profile likelihood Time to move on Encourage interaction with all physics groups ATLAS/CMS interaction focused on RooStats ( Kyle talk next)

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