Upper Limits and Discovery in Search for Exotic Physics

1. Upper Limits and Discovery in Search for Exotic Physics Jan Conrad Royal Institute of Technology (KTH) Stockholm Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20061

2. Outline • Discovery • Confidence Intervals • The problem of nuisance parameters (“systematic uncertainties”) • Averaging • Profiling • Analysis optimization • Summary Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20062

3. General approach to claiming discovery (hypothesis testing) • Assume an alleged physics process characterized by a signal parameter s (flux of WIMPS, Micro Blackholes .... etc.) • One can claim discovery of this process if the observed data is very unlikely to come from the null hypothesis , H0, being defined as non-existence of this process (s=0). ”Very unlikely” is hereby quantified as the ”signifcance” probability αsign, taken to be a small number (often 5 σ ~ 10-7). • Mathematically this is done by comparing the p-value with αsign and reject H0 if p –value < αsign Actually observed value of the test statistics test statistics, T, could be for example χ2 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20063

4. P-values and the Neyman Pearson lemma • Uniformly most powerful test statistic is the likelihood ratio : • For p-values, we need to know the null-distribution of T. Therefore it comes handy that asymptotically: • Often it is simply assumed that the null-distribution is χ2 but be careful ! see e.g. J.C. , presented at NuFACT06, Irvine, USA, Aug. 2006 L. Demortier, presented at BIRS, Banff, Canada, July 2006 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20064

5. Type I, type II error and power • Type I error:Reject H0, though it is true. Prob(Type I error) = α • Type II error:Accept H0, though it is false • Power:1 - β = 1 – Prob(Type II error) In words: given H1, what is the probability that we will reject H0 at given significance α ? In other words: what is the probability that we detect H1 ? • In designing a test, you want correct Type I error rate (this controls the number of false detections) and as large power as possible . Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20065

6. Why 5 s? • … traditional: we have seen 3 s significances disappear (….we also have seen 5 s signficances disappear on the other hand ….) • Principal reasoning (here done for the LHC): • LHC searches: 500 searches each of which has 100 resolution elements (mass, angle bins, etc.)  5 x 104 chances to find something. • One experiment: False positive rate at 5 s(5 x 104) (3 x 10-7) = 0.015. OK ! • Two experiments: • Assume we want to produce < 100 unneccessary theory papers •  allowable false positive rate: 10. •  2 (5 x 104) (1 x 10-4) = 10  3.7 s required. • Required other experiment verification: (1 x 10-3)(10) = 0.01  3.1 s required. It seems that the same reasoning would lead to smaller required signficance probabilities for EP searches in NT. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20066

7. Confidence Intervals (CI) • 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 %): • Bayesian: • Frequentist: • 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 • Confidence intervals are often used for hypothesis testing. G. Feldman & R. Cousins, Phys. RevD57:3873-3889 See e.g. J.C. presented at NuFACT06, Irvine, USA, Aug. 2006 K. S. Cranmer, PhyStat 2005, Oxford, Sept. 2005 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20067

8. Nuisance parameters1) • 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) • 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 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20068

9. How to ”get rid” of the nuisance parameters ? • There is a wealth of approaches to dealing with nuisance parameters. Two are particularly common: • Averaging (either the likelihood or the PDF): • Profiling (either the likelihood or the PDF): • ... less common, but correct per construction: fully frequentist, see e.g: Bayesian G. Punzi, PHYSTAT 2005, Oxford, Sept. 2005 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 20069

10. NT searches for EP: why things are bad ..... and good. • Bad • Low statistics makes the use of asymptotic methods doubtful • systematic uncertainties are large. • Good: • Many NT analyses are single channel searches with relatively few nuisance parameters •  rigorous methods are computationally feasible (even fully frequentist) Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200610

11. 1 -α 1 over-covering 0.9 under-covering s Coverage • 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). • Coverage is a necessary and sufficient condition for a valid CI calculation method Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200611

12. Averaging: hybrid Bayesian confidence intervals • Example PDF: • Perform Neyman-Construction with this new PDF (we will assume Feldman & Cousins in the remainder of this talk) • Treats nuisance parameters Bayesian, but performs a frequentist construction. Integral is performed in true variables  Bayesian J.C, O. Botner, A. Hallgren, C. de los Heros Phys. RevD67:012002,2003 R. Cousins & V. Highland Nucl. Inst. Meth. A320:331-335,1992 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200612

13. Coverage of hybrid method. Use Log-normal if large uncertainties !!!!! (1- α)MC true s true s F.Tegenfeldt & J.C. Nucl. Instr. Meth.A539:407-413, 2005 J.C & F. Tegenfeldt , PhyStat 05, Oxford, Sept. 2005, physics/0511055 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200613

14. Commercial break: pole++ • Bayesian treatment in FC ordering Neyman construction • treats P(n|εs +b) • Consists of C++ classes: • Pole calculate limits • Coverage coverage studies • Combine combine experiments • Nuisance parameters • supports flat, log-normal and Gaussian uncertainties in efficiency and background • Correlations (multi-variate distributions and uncorrelated case) • Code and documentation available from: • http://cern.ch/tegen/statistics.html J.C & F. Tegenfeldt , Proceedings PhyStat 05, physics/0511055 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200614

15. Example: hybrid Bayesian in NTs • From Daan Huberts talk (this conference): with systematicswithout systematics Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200615

16. Profiling: Profile Likelihood confidence intervals meas n, meas. b MLE of b given s MLE of b and s given observations 2.706 To extract limits: Lower limit Upper Limit Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200616

17. From MINUIT manual • 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 ΔΧ2 = 2.71 (90%), ΔΧ2 = 1.07 (70 %) Profile Likelihood Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200617

18. Background: Poisson (unc ~ 20 % -- 40 %) , Efficiency: binomial (unc ~ 12%) Rolke et al Minuit Coverage of profile likelihood Available as TRolke in ROOT ! Should be able to treat common NT cases (1- α)MC W. Rolke, A. Lopez, J.C. Nucl. Inst.Meth A 551 (2005) 493-503 true s Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200618

19. Profile likelihood goes LHC. • Basic idea: calculate 5 σ confidence interval and claim discovery if s = 0 is not included. • Straw-man model: • Typical: b = 100, т = 1 ( 10 % sys. Uncertainty on b) Size of side band region - 35 events!! - 17 events!! K. S. Cranmer, PHYSTAT 2005, Oxford, Sept. 2005 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200619

20. Analysis optimisation • Consider some cut-value t. Analysis is optimised defining a figure of merit (FOM). Very common: • Alternatively, optimize for most stringent upper limit. The corresponding figure of merit is the model rejection factor, MRF: Mean upper limit (only bg) G. Hill & K. Rawlins, Astropart. Phys. 19:393-402,2003 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200620

21. In case of systematics ? • Simplest generalizations one could think of: • In general, I do not think it makes a difference unless: NO ! Yes ! Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200621

22. Optimisation for discovery and upper limit at the same time ? • Fix significance (e.g αsign = 5 σ) and confidence level (e.g. 1-αCL = 99 %). Then define sensitivity region in s by : • The FOM can be defined to optimize this quantity (e.g simple counting experiment): Signal efficiency Number of σ (here assumed αsign = 1 – αCL) G. Punzi, PHYSTAT 2003, SLAC, Aug. 2003 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200622

23. Conclusions/Final Remarks • Two methods to calculate CI and claim discovery in presence of ”systematic” uncertainties have been discussed. • The methods presented here are certainly suitable for searches for Exotic Physics with Neutrino Telescopes and code exists which works ”out of the box” • Remark: the ”simplicity” of the problem (single channel, small number of nuisance parameters) make even rigorous methods applicable • Remark 2: the LHC example shows that for large signficances (discovery) hybrid Bayesian might be problematic. • I discussed briefly the issue of sensitivity and analysis optimisation. Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200623

24. Backup Slides Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200624

25. B0s µ+µ- J.C & F. Tegenfeldt , Proceedings PhyStat 05, physics/0511055 Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200625

26. Neyman construction Exp 3 Exp 2 Exp 1 One additional degree of freedom: ORDER in which you inlcude the n into the belt J. Neyman, Phil. Trans. Roy. Soc. London A, 333, (1937) Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200626

27.  s Projection method with appropriate ordering. Ordering function: (Punzi, PhyStat05) Poisson signal, Gauss eff. Unc (10 %) Can be any ordering in prime observable sub-space, in this case Likelihood ratio (Feldman & Cousins) ~ FC Profile Average coverage Max/Min coverage s Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200627

28. FC ordering: coverage (1- α)MC Calculated by Pseudo-experiments Nominal coverage true s Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200628

29. Some methods for p-value calculation • Conditioning • Prior-predictive • Posterior-predictive • Plug-In • Likelihood Ratio • Confidence Interval • Generalized frequentist Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200629

30. Some methods for confidence interval calculation (the Banff list) • Bayesian • Feldman & Cousins with Bayesian treatment of nuisance parameters (Hybrid Bayesian) • Profile Likelihood • Modified Likelihood • Feldman & Cousins with Profile Likelihood • Fully frequentist • Empirical Bayes Jan Conrad (KTH, Sthlm) EPNT 06 20 September 200630