1. 1 LIMITS Why limits? Methods for upper limits Desirable properties Dealing with systematics Feldman-Cousins Recommendations

2. 2 WHY LIMITS? Michelson-Morley experiment  death of aether HEP experiments CERN CLW (Jan 2000) FNAL CLW (March 2000) Heinrich, PHYSTAT-LHC, “Review of Banff Challenge”

3. 3 SIMPLE PROBLEM? Gaussian ~ exp{-0.5*(x-μ) 2 /σ 2 } No restriction on μ, σ known exactly μ  x 0 + k σ BUT Poisson {μ = sε + b} s ≥ 0 ε and b with uncertainties Not like : 2 + 3 = ? N.B. Actual limit from experiment = Expected (median) limit

4. 4 Bayes (needs priors e.g. const, 1/μ, 1/ √ μ, μ, …..) Frequentist (needs ordering rule, possible empty intervals, F-C) Likelihood (DON’T integrate your L) χ 2 ( σ 2 =μ) χ 2 ( σ 2 = n) Recommendation 7 from CERN CLW: “Show your L” 1) Not always practical 2) Not sufficient for frequentist methods Methods (no systematics)

5. 5 Bayesian posterior  intervals Upper limit Lower limit Central interval Shortest

6. 6 90% C.L. Upper Limits x  x0x0

7. 7 Ilya Narsky, FNAL CLW 2000

8. 8 DESIRABLE PROPERTIES Coverage Interval length Behaviour when n < b Limit increases as σ b increases

9. 9 Δln L = -1/2 rule If L (μ) is Gaussian, following definitions of σ are equivalent: 1) RMS of L ( µ ) 2) 1/√(-d 2 L /d µ 2 ) 3) ln( L (μ±σ) = ln( L (μ 0 )) -1/2 If L (μ) is non-Gaussian, these are no longer the same “ Procedure 3) above still gives interval that contains the true value of parameter μ with 68% probability ” Heinrich: CDF note 6438 (see CDF Statistics Committee Web-page) Barlow: Phystat05

10. 10 COVERAGE How often does quoted range for parameter include param’s true value? N.B. Coverage is a property of METHOD, not of a particular exptl result Coverage can vary with Study coverage of different methods of Poisson parameter, from observation of number of events n Hope for: Nominal value 100%

11. 11 COVERAGE If true for all : “correct coverage” P< for some “undercoverage” (this is serious !) P> for some “overcoverage” Conservative Loss of rejection power

12. 12 Coverage : L approach (Not frequentist) P(n, μ) = e -μ μ n /n! (Joel Heinrich CDF note 6438) -2 lnλ< 1 λ = P(n,μ)/P(n,μ best ) UNDERCOVERS

13. 13 Frequentist central intervals, NEVER undercovers (Conservative at both ends)

14. 14 Feldman-Cousins Unified intervals Frequentist, so NEVER undercovers

15. 15 Probability ordering Frequentist, so NEVER undercovers

16. 16  = (n- µ) 2 /µ Δ = 0.1 24.8% coverage? NOT frequentist : Coverage = 0%  100%

17. 17 COVERAGE N.B. Coverage alone is not sufficient e.g. Clifford (CERN CLW, 2000) “Friend thinks of number Procedure for providing interval that includes number 90% of time.”

18. 18 COVERAGE N.B. Coverage alone is not sufficient e.g. Clifford (CERN CLW, 2000) Friend thinks of number Procedure for providing interval that includes number 90% of time. 90%: Interval = -  to +  10%: number = 102.84590135…..

19. 19 INTERVAL LENGTH Empty  Unhappy physicists Very short  False impression of sensitivity Too long  loss of power (2-sided intervals are more complicated because ‘shorter’ is not metric- independent: e.g. 0  4 or 4  9)

20. 20 90% Classical interval for Gaussian σ = 1 μ ≥ 0 e.g. m 2 (ν e )

21. 21 Behaviour when n < b Frequentist: Empty for n < < b Frequentist: Decreases as n decreases below b Bayes: For n = 0, limit independent of b Sen and Woodroofe: Limit increases as data decreases below expectation

22. 22 FELDMAN - COUSINS Wants to avoid empty classical intervals  Uses “ L -ratio ordering principle” to resolve ambiguity about “which 90% region?”  [Neyman + Pearson say L -ratio is best for hypothesis testing] Unified  No ‘Flip-Flop’ problem

23. 23 X obs = -2 now gives upper limit

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26. 26 Black lines Classical 90% central interval Red dashed: Classical 90% upper limit Flip-flop

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28. 28 Poisson confidence intervals. Background = 3 Standard Frequentist Feldman - Cousins

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42. 42 Recommendations? CDF note 7739 (May 2005) Decide method in advance No valid method is ruled out Bayes is simplest for incorporating nuisance params Check robustness Quote coverage Quote sensitivity Use same method as other similar expts Explain method used

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45. 45 Caltech Workshop, Feb 11th

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55. 55 Tomorrow is last day of this visit Contact me at: l.lyons@physics.ox.ac.uk

56. 56 Peasant and Dog 1)Dog d has 50% probability of being 100 m. of Peasant p 2)Peasant p has 50% probability of being within 100m of Dog d dp x River x =0River x =1 km

57. 57 Given that: a) Dog d has 50% probability of being 100 m. of Peasant, is it true that: b ) Peasant p has 50% probability of being within 100m of Dog d ? Additional information Rivers at zero & 1 km. Peasant cannot cross them. Dog can swim across river - Statement a) still true If Dog at –101 m, Peasant cannot be within 100m of Dog Statement b) untrue