1. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 1 Upper Limits and Discovery in Search for Exotic Physics Jan Conrad Royal Institute of Technology (KTH) Stockholm

2. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 2 Outline l Discovery l Confidence Intervals l The problem of nuisance parameters (“systematic uncertainties”) l Averaging l Profiling l Analysis optimization l Summary

3. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 3 General approach to claiming discovery (hypothesis testing) l Assume an alleged physics process characterized by a signal parameter s (flux of WIMPS, Micro Blackholes.... etc.) l One can claim discovery of this process if the observed data is very unlikely to come from the null hypothesis, H 0, being defined as non- existence of this process (s=0). ”Very unlikely” is hereby quantified as the ”signifcance” probability α sign, taken to be a small number (often 5 σ ~ 10 -7 ). l Mathematically this is done by comparing the p-value with α sign and reject H 0 if p –value < α sign test statistics, T, could be for example χ 2 Actually observed value of the test statistics

4. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 4 P-values and the Neyman Pearson lemma l Uniformly most powerful test statistic is the likelihood ratio : l For p-values, we need to know the null-distribution of T. Therefore it comes handy that asymptotically: l Often it is simply assumed that the null-distribution is χ 2 but be careful ! see e.g. J.C., presented at NuFACT06, Irvine, USA, Aug. 2006 L. Demortier, presented at BIRS, Banff, Canada, July 2006

5. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 5 Type I, type II error and power l Type I error: Reject H 0, though it is true. Prob(Type I error) = α l Type II error: Accept H 0, though it is false l Power: 1 - β = 1 – Prob(Type II error) In words: given H 1, what is the probability that we will reject H 0 at given significance α ? In other words: what is the probability that we detect H 1 ? l In designing a test, you want correct Type I error rate (this controls the number of false detections) and as large power as possible.

6. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 6 Why 5 ? … traditional: we have seen 3  significances disappear (….we also have seen 5  signficances disappear on the other hand ….) l Principal reasoning (here done for the LHC): l LHC searches: 500 searches each of which has 100 resolution elements (mass, angle bins, etc.)  5 x 10 4 chances to find something. One experiment: False positive rate at 5   (5 x 10 4 ) (3 x 10 -7 ) = 0.015. OK ! l Two experiments: Assume we want to produce < 100 unneccessary theory papers  allowable false positive rate: 10.   2 (5 x 10 4 ) (1 x 10 -4 ) = 10  3.7  required.  Required other experiment verification: (1 x 10 -3 )(10) = 0.01  3.1  required. It seems that the same reasoning would lead to smaller required signficance probabilities for EP searches in NT.

7. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 7 Confidence Intervals (CI) l Instead of doing a hypothesis test, we might want to do a interval estimate on the parameter s with confidence level 100(1 – α) % (e.g. 90 %): l Bayesian: l Frequentist: l Invert by e.g. Neyman construction of confidence intervals (no time to explain) - special case 1: n 2 =   upper limit - special case 2: two sided/one sided limits depending on observation  Feldman & Cousins l Confidence intervals are often used for hypothesis testing. G. Feldman & R. Cousins, Phys. Rev D57:3873-3889 See e.g. J.C. presented at NuFACT06, Irvine, USA, Aug. 2006 K. S. Cranmer, PhyStat 2005, Oxford, Sept. 2005

8. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 8 Nuisance parameters 1) l Nuisance parameters are parameters which enter the data model, but which are not of prime interest (expected background, estimated signal/background efficiencies etc. pp., often called systematic uncertainties) l You don’t want to give CIs (or p-values) dependent on nuisance parameters  need a way to get rid of them 1) Applies to both confidence intervals and nuisance parameters

9. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 9 How to ”get rid” of the nuisance parameters ? l There is a wealth of approaches to dealing with nuisance parameters. Two are particularly common: l Averaging (either the likelihood or the PDF): l Profiling (either the likelihood or the PDF): l... less common, but correct per construction: fully frequentist, see e.g: Bayesian G. Punzi, PHYSTAT 2005, Oxford, Sept. 2005

10. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 10 NT searches for EP: why things are bad..... and good. l Bad l Low statistics makes the use of asymptotic methods doubtful l systematic uncertainties are large. l Good: l Many NT analyses are single channel searches with relatively few nuisance parameters  rigorous methods are computationally feasible (even fully frequentist)

11. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 11 Coverage l A method is said to have coverage (1-α) if, in infinitely many repeated experiments the resulting CIs include (cover) the true value in a fraction (1-α) of all cases (irrespective of what the true value is). l Coverage is a necessary and sufficient condition for a valid CI calculation method 1 -α s 1 0.9 over-covering under-covering

12. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 12 Averaging: hybrid Bayesian confidence intervals J.C, O. Botner, A. Hallgren, C. de los Heros Phys. Rev D67:012002,2003 R. Cousins & V. Highland Nucl. Inst. Meth. A320:331-335,1992 l Example PDF: l Perform Neyman-Construction with this new PDF (we will assume Feldman & Cousins in the remainder of this talk) l Treats nuisance parameters Bayesian, but performs a frequentist construction. Integral is performed in true variables  Bayesian

13. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 13 Coverage of hybrid method. J.C & F. Tegenfeldt, PhyStat 05, Oxford, Sept. 2005, physics/0511055 (1- α) MC true s F.Tegenfeldt & J.C. Nucl. Instr. Meth.A539:407-413, 2005 Use Log-normal if large uncertainties !!!!!

14. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 14 Commercial break: pole++ l Bayesian treatment in FC ordering Neyman construction l treats P(n|εs +b) l Consists of C++ classes: l Pole calculate limits l Coverage coverage studies l Combinecombine experiments l Nuisance parameters l supports flat, log-normal and Gaussian uncertainties in efficiency and background l Correlations (multi-variate distributions and uncorrelated case) l Code and documentation available from: l http://cern.ch/tegen/statistics.html http://cern.ch/tegen/statistics.html J.C & F. Tegenfeldt, Proceedings PhyStat 05, physics/0511055

15. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 15 Example: hybrid Bayesian in NTs l From Daan Huberts talk (this conference): with systematics without systematics

16. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 16 Profiling: Profile Likelihood confidence intervals Lower limitUpper Limit 2.706 meas n, meas. b MLE of b given s MLE of b and s given observations To extract limits:

17. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 17 From MINUIT manual l See F. James, MINUIT Reference Manual, CERN Library Long Write-up D506, p.5: “ The MINOS error for a given parameter is defined as the change in the value of the parameter that causes the F’ to increase by the amount UP, where F’ is the minimum w.r.t to all other free parameters”. Confidence Interval Profile LikelihoodΔΧ 2 = 2.71 (90%), ΔΧ 2 = 1.07 (70 %)

18. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 18 Coverage of profile likelihood Background: Poisson (unc ~ 20 % -- 40 %), Efficiency: binomial (unc ~ 12%) Rolke et al Minuit W. Rolke, A. Lopez, J.C. Nucl. Inst.Meth A 551 (2005) 493- 503 (1- α) MC true s Available as TRolke in ROOT ! Should be able to treat common NT cases

19. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 19 Profile likelihood goes LHC. l Basic idea: calculate 5 σ confidence interval and claim discovery if s = 0 is not included. l Straw-man model: l Typical: b = 100, т = 1 (  10 % sys. Uncertainty on b) K. S. Cranmer, PHYSTAT 2005, Oxford, Sept. 2005 Method Calculated coverage (nom. 5σ) Critical number of events (b obs = 100) Ignore ”systematics”3.1150 Bayesian treatment4.2178 Profile likelihood5.0185 Size of side band region - 17 events!! - 35 events!!

20. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 20 Analysis optimisation l Consider some cut-value t. Analysis is optimised defining a figure of merit (FOM). Very common: l Alternatively, optimize for most stringent upper limit. The corresponding figure of merit is the model rejection factor, MRF: G. Hill & K. Rawlins, Astropart. Phys. 19:393-402,2003 Mean upper limit (only bg)

21. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 21 In case of systematics ? l Simplest generalizations one could think of: l In general, I do not think it makes a difference unless: NO ! Yes !

22. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 22 Optimisation for discovery and upper limit at the same time ? l Fix significance (e.g α sign = 5 σ) and confidence level (e.g. 1-α CL = 99 %). Then define sensitivity region in s by : l The FOM can be defined to optimize this quantity (e.g simple counting experiment): G. Punzi, PHYSTAT 2003, SLAC, Aug. 2003 Signal efficiency Number of σ (here assumed α sign = 1 – α CL )

23. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 23 Conclusions/Final Remarks l Two methods to calculate CI and claim discovery in presence of ”systematic” uncertainties have been discussed. l The methods presented here are certainly suitable for searches for Exotic Physics with Neutrino Telescopes and code exists which works ”out of the box” l Remark: the ”simplicity” of the problem (single channel, small number of nuisance parameters) make even rigorous methods applicable l Remark 2: the LHC example shows that for large signficances (discovery) hybrid Bayesian might be problematic. l I discussed briefly the issue of sensitivity and analysis optimisation.

24. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 24 l Backup Slides

25. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 25 B 0 s  µ+µ- CDF1 CDF 2 Eff uncertainty [%]18.2 16.0 Eff uncertainty [%]20.3 19.2 Corr. eff. Uncertainty [%] 15.5 95 % CI [10- 7 ]2.54.3 95 % combined limit [10- 7 ] 1.7 (2.0 in Bernhard et. al.) J.C & F. Tegenfeldt, Proceedings PhyStat 05, physics/0511055

26. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 26 Neyman construction J. Neyman, Phil. Trans. Roy. Soc. London A, 333, (1937) Exp 1 Exp 2 Exp 3 One additional degree of freedom: ORDER in which you inlcude the n into the belt

27. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 27 Average coverage s  Max/Min coverage Ordering function: (Punzi, PhyStat05) Can be any ordering in prime observable sub- space, in this case Likelihood ratio (Feldman & Cousins) ~ FC Profile s Poisson signal, Gauss eff. Unc (10 %) Projection method with appropriate ordering.

28. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 28 FC ordering: coverage (1- α) MC true s Nominal coverage Calculated by Pseudo- experiments

29. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 29 Some methods for p-value calculation l Conditioning l Prior-predictive l Posterior-predictive l Plug-In l Likelihood Ratio l Confidence Interval l Generalized frequentist

30. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 30 Some methods for confidence interval calculation (the Banff list) l Bayesian l Feldman & Cousins with Bayesian treatment of nuisance parameters (Hybrid Bayesian) l Profile Likelihood l Modified Likelihood l Feldman & Cousins with Profile Likelihood l Fully frequentist l Empirical Bayes