1. Everything You Always Wanted To Know About Limits* Roger Barlow Manchester University YETI06 *But were afraid to ask

2. Roger Barlow: YETI06Everything you wanted to know about limits Slide 2 Summary Prediction confronts data & sees small/zero signal Frequentist probability and Confidence Level language Bayesian Probability (Health Warning) Gaussian  ln L= -½ Zero events Few events: Confidence belt The horrendous case of large backgrounds Extension to several parameters Likelihood

3. Roger Barlow: YETI06Everything you wanted to know about limits Slide 3 Model predictions Input model and parameters Low energy Lagrangian Feynman Rules for Feynman diagrams Cross Sections and Branching Ratios Experiment duration, luminosity, Efficiency etc Number of events Monte Carlo programs Cuts designed to bring out signal Data

4. Roger Barlow: YETI06Everything you wanted to know about limits Slide 4 What happens if there’s nothing there? Even if your analysis finds no events, this is still useful information about the way the universe is built Want to say more than: “We looked for X, we didn’t see it.” Need statistics – which can’t prove anything. “We show that X probably has a mass greater than../a coupling smaller than…”

5. Roger Barlow: YETI06Everything you wanted to know about limits Slide 5 Probability(1): Frequentist Define Probability of X as P(X)=Limit N  ∞ N(X) / N Examples: coins, dice, cards For continuous x extend to Probability Density P(x to x+dx)=p(x)dx Examples: Measuring continuous quantities (p(x) often Gaussian) Parton momentum fractions (proton pdfs)

6. Roger Barlow: YETI06Everything you wanted to know about limits Slide 6 Digression: likelihood Probability distribution of random variable x often depends on some parameter a. Joint function p(x,a) Considered as p(x)| a this is the pdf. Normalised: ∫p(x)dx=1 Considered as p(a)| x this is the Likelihood L(a) Not ‘likelihood of a’ but ‘likelihood that a would give x’ Not normalised. Indeed, must never be integrated.

7. Roger Barlow: YETI06Everything you wanted to know about limits Slide 7 Limitation of Frequentist Probability Have to say “The statement ‘It will rain tomorrow.’ is probably true.” Can then even quantify (meteorology). Can’t say “It will probably rain tomorrow.” There is only one tomorrow. P is either 1 or 0

8. Roger Barlow: YETI06Everything you wanted to know about limits Slide 8 Interpreting physics results: M t =173±2 GeV/c 2 Can’t say ‘M t has a 68% probability of lying between 171 and 175 GeV/c 2 ’ Have to say ‘The statement “M t lies between 171 and 175 GeV/c 2” has a 68% probability of being true’ i.e. if you always say a value lies within its error bars, you will be right 68% of the time. Say “M t lies between 171 and 175 GeV/c 2” with 68% Confidence. Or 169-177 with 95% confidence. Or…

9. Roger Barlow: YETI06Everything you wanted to know about limits Slide 9 Interpreting null result Your analysis searches for events. Sees none. Use Poisson formula: P(n; )=e - n /n! Small could well give 0 events =0.5 gives P(0)=61% =1.0 gives P(0)=37% =2.3 gives P(0)=10% =3.0 gives P(0)=5% If you always say ‘  3.0’ you will be right (at least) 95% of the time.  3.0 – with 95% confidence (a.k.a 5% significance.) ‘If is actually 3, or more, the probability of a fluctuation as far as zero is only 5%, or less.’ given by model parameters. Limit on translates to limit on mass, coupling,,branching ratio or whatever

10. Roger Barlow: YETI06Everything you wanted to know about limits Slide 10 Probability(2): Bayesian P(X) expresses by degree of belief in X Can calibrate against cards, dice, etc. Extend to probability density p(x) as before No restrictions on X or x. Rain, M T, M H, whatever Interpret physics results using Bayes’ Theorem: p posterior (a|data)  p(data|a) x p prior (a)

11. Roger Barlow: YETI06Everything you wanted to know about limits Slide 11 Bayes at work = x P(0 events| ) (Likelihood) Prior: uniform Posterior P( )  3 P( ) d = 0.95 0 Same as Frequentist limit - Happy coincidence Zero events seen

12. Roger Barlow: YETI06Everything you wanted to know about limits Slide 12 Bayes at work again = x P(0 events| ) Prior: uniform in ln Posterior P( )  3 P( ) d >> 0.95 0 Is that uniform prior really credible? Upper limit totally different!

13. Roger Barlow: YETI06Everything you wanted to know about limits Slide 13 Bayes: the bad news The prior affects the posterior. It is your choice. That makes the measurement subjective. This is BAD. (We’re physicists, dammit!) A Uniform Prior does not get you out of this. SUSY ‘parameter space’ is not a ‘phase space’ Attempts to invent universally-agreed priors (‘Objective’ and/or ‘Reference’ Priors) have not worked Better news: If there is a lot of data then the prejudicial effects of the choice of prior can be small. This should ALWAYS be checked for (‘robustness under choice of prior’.)

14. Roger Barlow: YETI06Everything you wanted to know about limits Slide 14 Frequentist versus Bayesian? Statisticians do a lot of work with Bayesian statistics and there are a lot of useful ideas. But they are careful about checking for robustness under choice of prior. Beware snake-oil merchants in the physics community who will sell you Bayesian statistics (new – cool – easy – intuitive) and don’t bother about robustness. Use Frequentist methods when you can and Bayesian when you can’t (and check for robustness.) But ALWAYS be aware which you are using.

15. Roger Barlow: YETI06Everything you wanted to know about limits Slide 15 A Gaussian Measurement No problems p(x)=exp[-(x-  ) 2 /2  2 ]/  √2  x:  symmetric x is within ±  of  with 68% probability  is within ±  of x at 68% confidence x is above  -1.645  with 95% probability  is below x+1.645  at 95% confidence Choice of confidence level and arrangement Can read regions off log likelihood plot as L(a)=exp[-(x-  ) 2 /2  2 ]/  √2  Ln L  -(  -x) 2 /2  2 68% region corresponds to fall of ½ from peak

16. Roger Barlow: YETI06Everything you wanted to know about limits Slide 16 A Poisson measurement You detect 5 events. Best value 5. But what about the errors? 1.5±√5=5±2.24 Assumes e - n /n! is Gaussian in n. True only for large - and 5 is small 2. Find points where log likelihood falls by ½. Assumes e - n /n! is Gaussian in. Gives upper error of 2.58, lower error of 1.92

17. Roger Barlow: YETI06Everything you wanted to know about limits Slide 17 3: Doing it properly: Confidence belt (Neyman interval) Use e - n /n! For any true the probability that (n, ) is within the belt is 68% (or more) by construction For any n, lies in [ -, + ] at 68% confidence Get upper error 3.38, lower error 2.16 n - + 68% 16% Technique works for any CL, and single or double sided

18. Roger Barlow: YETI06Everything you wanted to know about limits Slide 18 Consumer guide  ln L =- ½ is a standard and easy to use. Fine for everyday use. (Though for a simple count the Neyman limit is quite easy) For 90% 1-sided (upper) limit use  ln L =-0.82 (1.28  ) For 95% use  ln L =-1.35 (1.645  ) Just plot the likelihood and read off the value. Then translate back to model parameters

19. Roger Barlow: YETI06Everything you wanted to know about limits Slide 19 Frequency method: the big problem Observe 5 events. Expected background of 0.9 events. Data = signal + background Say with 68% confidence: data in range 2.84 to 8.38 So say with 68% confidence: signal in range 1.94 to 7.48 Suppose expected* background 4.9. Or 6.9. Or 10.9 ? “We say at 68% confidence that the number of signal events lies between -8.06 and -2.52” This is technically correct. We are allowed to be wrong 32% of the time. But stupid. We know that the background happens to have a downward fluctuation but have no way of incorporating that knowledge *We assume that the background is calculated correctly

20. Roger Barlow: YETI06Everything you wanted to know about limits Slide 20 Strategy 1: Bayes Prior is uniform for positive, zero for negative. No problem. Get requirement (for n observed, known background b, 90% upper limit) 0.1=  n exp(- + -b) ( + +b) r /r!  n exp(-b) b r /r! Known as “the old PDG formula” or “Helene’s formula” or “that heap of crap”

21. Roger Barlow: YETI06Everything you wanted to know about limits Slide 21 Strategy 2: Feldman-Cousins Also called* ‘the Unified Approach’ Real physicists wait to see their result and then decide whether to quote an upper limit or a range. This ‘flip-flopping’ invalidates the method. They provide a procedure that incorporates it automatically, and always gives non-stupid results. Critics say (1) can lead to experiments quoting a range when they’re not claiming a discovery (2) is computationally intensive and (3) For zero observed events, the higher the background estimate the better (i.e. lower) the limit on signal * By Feldman and Cousins, principally

22. Roger Barlow: YETI06Everything you wanted to know about limits Slide 22 Strategy 3: CL s As used by LEP Higgs working group Generalisation of Helene formula Some quantity Q. Could be number of events, or something more clever CL b =P(Q or less|b) CL s+b =P(Q or less|s+b) CL s =CL s+b /CL b Used as confidence level. Optimise strategy using it and quote results

23. Roger Barlow: YETI06Everything you wanted to know about limits Slide 23 2(+) parameters Fix b, find 68% confidence range for a, using  ln L=-½ Fix a, find 68% range for b Combination (square) has 0.68 2 =46% a b L(a,b)  ln L=-½ circle has 39% Confidence Define regions through contours of log L – Confidence content given by 2  ln L=    for which P(    n)=CL Caution! Cannot redefine a as b+c+d, claim 3 parameters and cut with P(    3) instead of P(    1)

24. Roger Barlow: YETI06Everything you wanted to know about limits Slide 24 Summary Prediction confronts data & sees small/zero signal Frequentist probability and Confidence Level language Bayesian Probability (Health Warning) Gaussian  ln L= -½ Zero events Few events: Confidence belt The horrendous case of large backgrounds Extension to several parameters Likelihood

25. Roger Barlow: YETI06Everything you wanted to know about limits Slide 25 Remember! Zero events = 95% CL upper limit of 3 events If it’s more involved, plot the likelihood function and use  ln L=-½ for 68% central, etc Be suspicious of anything you don’t understand If you’re integrating the likelihood you are a Bayesian. I hope you know what you’re doing.

26. Roger Barlow: YETI06Everything you wanted to know about limits Slide 26 Further Reading Workshop on Confidence Limits, CERN yellow report 2000-005 Proc. Conf. Advanced Statistical Techniques in Particle Physics, Durham, IPPP02/39 Proc. PHYSTAT03 – SLAC-R-703 Proc PHYSTAT05, Oxford - forthcoming