1. Results from Small Numbers Roger Barlow Manchester 26 th September 2007

2. Results from Small Numbers2 Summary: 3 problems 1. Number of events small (including 0) 2. Number of events  Background 3. Uncertainties in background and efficiency

3. Results from Small Numbers3 Preamble: Particle Physics is about counting Pretty much everything is Poisson Statistics  N Numbers of events give cross sections, branching ratios… E.g. Branching ratio for some rare B decay Have T=200M B mesons, Efficiency E=0.02 Observe 100 events. Error  100=10 BR=N/TE BR=(25.0  2.5) 10 -6

4. Results from Small Numbers4 1. What do you do with zero? Observe 0 events. (Many searches do) BR=0  0 is obviously wrong We know BR is small. But not that it’s exactly 0. Combination of small value+bad luck can give 0 events Need to go back two steps to consider  what we mean by measurement errors  what we mean by probability There are two ways of doing this…

5. Results from Small Numbers5 Probability (conventional definition) Limit of frequency P(A)= Limit N  N(A)/N Standard (frequentist) definition. Two tricky features 1. P(A) depends not just on A but on the ensemble. Several ensembles may be possible 2. If you cannot define an Ensemble there is no such thing as probability Ensemble of Everything A

6. 6 Feature 1:There can be many Ensembles Probabilities belong to the event and the ensemble Insurance company data shows P(death) for 40 year old male clients = 1.4% (Classic example due to von Mises) Does this mean a particular 40 year old German has a 98.6% chance of reaching his 41 st Birthday? No. He belongs to many ensembles  German insured males  German males  Insured nonsmoking vegetarians  German insured male racing drivers  … Each of these gives a different number. All equally valid.

7. 7 Feature 2: Unique events have no ensemble Some events are unique. Consider “It will probably rain tomorrow.” There is only one tomorrow (Thursday 27 th September). There is NO ensemble. P(rain) is either 0/1 =0 or 1/1 = 1 Strict frequentists cannot say 'It will probably rain tomorrow'. This presents severe social problems.

8. 8 Circumventing the limitation A frequentist can say: “The statement ‘It will rain tomorrow’ has a 70% probability of being true.” by assembling an ensemble of statements and ascertaining that 70% (say) are true. (E.g. Weather forecasts with a verified track record) Say “It will rain tomorrow” with 70% confidence For unique events, confidence level statements replace probability statements.

9. Results from Small Numbers9 What is a measurement? M T =174  3 GeV What does it mean? For true value  and standard deviation  the probability (density) for a result x is (for the usual Gaussian measurement) P( x ; ,  )=( 1 /  2  ) exp-[( x -  ) 2 /2  2 ] So is there a 68% probability that M T lies between 171 and 177 GeV? No. M T is unique. It is either in that range or outside. (Soon we will know.) For a given , the probability that x lies within  is 68% This does not mean that for a given x, the ‘inverse’ probability that  lies within  is 68% P( x ; ,  ) cannot be used as a probability for . (It is called the likelihood function for  given x.)

10. Results from Small Numbers10 What a measurement error says A Gaussian measurement gives a result within  1  of the true value in 68% of all cases. The statement “  [ x - , x +  ]” has a 68% probability of being true, using the ensemble of all such statements. We say “  [ x - , x +  ]”, or “M T lies between 169 and 179 GeV” with 68% confidence. Can also say “  [ x - 2 , x + 2  ]” @ 95% or “  [- , x +  ]” @ 84% or whatever

11. Results from Small Numbers11 Extension beyond simple Gaussian Choose construction (functions x 1 (  ), x 2 (  ) ) for which P( x  [ x 1 (  ), x 2 (  ) ])  CL for all  Given a measurement X, make statement  [  LO,  HI ] @ CL Where X= x 2 (  LO ), X= x 1 (  HI )

12. Results from Small Numbers12 Confidence Belt x  Example: proportional Gaussian  = 0.1  Measures with 10% accuracy Result (say) 100.0  LO =90.91  HI = 111.1 Constructed horizontally such that the probability of a result lying inside the belt is 68%(or whatever) Read vertically using the measurement X

13. Results from Small Numbers13 Use for small numbers Can choose CL Just use one curve to give upper limit Discrete observable makes smooth curves into ugly staircases Observe n. Quote upper limit as  HI from solving  0 n P(r,  HI ) =  0 n e -  HI  HI r /r! = 1-CL English translation. n is small.  can’t be very large. If the true value is  HI (or higher) then the chance of a result this small (or smaller) is only (1-CL) (or less)

14. 14 Poisson table Upper limits n90%95%99% 02.30 3.004.61 13.894.746.64 25.326.308.41 36.687.7510.05 47.999.1511.60 59.2710.5113.11.....

15. Results from Small Numbers15 Alternative: Bayesian (Subjective) Probability P(A) is a number describing my degree of belief in A 1=certain belief. 0=total disbelief Intermediate beliefs can be calibrated against simple frequentist probabilities. P(A)=0.5 means: I would be indifferent given the choice of betting on A or betting on a coin toss. A can be anything. Measurements, true values, rain, M T, M H, horse races, existence of God, age of the current king of France… Very adaptable. But no guarantee my P(A) is the same as your P(A). Subjective = unscientific?

16. Results from Small Numbers16 Bayes’ Theorem General (uncontroversial) form P(A|B)=P(B|A) P(A) P(B) “Bayesian” form P(Theory|Data)=P(Data|Theory) P(Theory) P(Data) PriorPosterior

17. Results from Small Numbers17 Bayes at work Prior and Posterior can be numbers Successful predictions boost belief in theory Several experiments modify belief cumulatively Prior and Posterior can be distributions P(  |x)  P(x|  ) P(  ) Ignore normalisation problems = X

18. 18 Example: Poisson P(r,  )=exp(-  )  r /r! With uniform prior this gives posterior for  Shown for various small r results Read off intervals... r=6 r=2 r=1 r=0 P(  ) 

19. 19 Upper limits Upper limit from n events  0  HI exp(-  )  n /n! d  = CL Repeated integration by parts:  0 n exp(-  HI )  HI r /r! = 1-CL Same as frequentist limit This is a coincidence! Lower Limit formula is not the same

20. 20 Result depends on Prior Example: 90% CL Limit from 0 events Prior flat in  Prior flat in  X X = = 1.65 2.30

21. Results from Small Numbers21 Health Warning Results using Bayesian Statistics will depend on the prior Choice of prior is arbitrary (almost always). ‘Uniform’ is not the answer. Uniform in what? Serious statistical analyses will try several priors and check how much the result shifts (robustness) Many physicists don’t bother

22. Results from Small Numbers22 2. Next problem: add a background  =S+b Frequentist Approach: 1. Find range for  2. Subtract b to get range for S Examples: See 5 events, background 1.2 95% Upper limit: 10.5  9.3  See 5 events, background 5.1 95% Upper limit: 10.5  5.2 ? See 5 events, background 10.6 95% Upper limit: 10.5  -0.1 

23. Results from Small Numbers23 S< -0.1? What’s going on? If N<b we know that there is a downward fluctuation in the background. (Which happens…) But there is no way of incorporating this information without messing up the ensemble Really strict frequentist procedure is to go ahead and publish. We know that 5% of 95%CL statements are wrong – this is one of them Suppressing this publication will bias the global results

24. Results from Small Numbers24  =S+b for Bayesians No problem! Prior for  is uniform for S  b Multiply and normalise as before Posterior Likelihood Prior Read off Confidence Levels by integrating posterior = X

25. 25 Incorporating Constraints: Poisson Work with total source strength (s+b) you know is greater than the background b Need to solve Formula not as obvious as it looks.

26. 26 Feldman Cousins Method Works by attacking what looks like a different problem... * by Feldman and Cousins, mostly

27. 27 Feldman Cousins:  =s+b b is known. N is measured. s is what we're after This is called 'flip-flopping' and BAD because is wrecks the whole design of the Confidence Belt Suggested solution: 1) Construct belts at chosen CL as before 2) Find new ranking strategy to determine what's inside and what's outside 1 sided 90% 2 sided 90%

28. 28 Feldman Cousins: Ranking First idea (almost right) Sum/integrate over range of (s+b) values with highest probabilities for this observed N. (advantage that this is the shortest interval) Glitch: Suppose N small. (low fluctuation) P(N;s+b) will be small for any s and never get counted Instead: compare to 'best' probability for this N, at s=N-b or s=0 and rank on that number Such a plot does an automatic ‘flip-flop’ N~b single sided limit (upper bound) for s N>>b 2 sided limits for s

29. 29 How it works Has to be computed for the appropriate value of background b. (Sounds complicated, but there is lots of software around) As n increases, flips from 1- sided to 2-sided limits – but in such a way that the probability of being in the belt is preserved s n Means that sensible 1-sided limits are quoted instead of nonsensical 2- sided limits!

30. 30 Arguments against using Feldman Cousins Argument 1 It takes control out of hands of physicist. You might want to quote a 2 sided limit for an expected process, an upper limit for something weird Counter argument: This is the virtue of the method. This control invalidates the conventional technique. The physicist can use their discretion over the CL. In rare cases it is permissible to say ”We set a 2 sided limit, but we're not claiming a signal”

31. 31 Feldman Cousins: Argument 2 Argument 2 If zero events are observed by two experiments, the one with the higher background b will quote the lower limit. This is unfair to hardworking physicists Counterargument An experiment with higher background has to be ‘lucky’ to get zero events. Luckier experiments will always quote better limits. Averaging over luck, lower values of b get lower limits to report. Example: you reward a good student with a lottery ticket which has a 10% chance of winning $10. A moderate student gets a ticket with a 1% chance of winning $20. They both win. Were you unfair? Example: you reward a good student with a lottery ticket which has a 10% chance of winning £ 10. A moderate student gets a ticket with a 1% chance of winning £ 20. They both win. Were you unfair?

32. Results from Small Numbers32 3. Including Systematic Errors  =aS+b  is predicted number of events S is (unknown) signal source strength. Probably a cross section or branching ratio or decay rate a is an acceptance/luminosity factor known with some (systematic) error b is the background rate, known with some (systematic) error

33. Results from Small Numbers33 3.1 Full Bayesian Assume priors for S (uniform?) For a (Gaussian?) For b (Poisson or Gaussian?) Write down the posterior P(S,a,b). Integrate over all a,b to get marginalised P(s) Read off desired limits by integration

34. Results from Small Numbers34 3.2 Hybrid Bayesian Assume priors For a (Gaussian?) For b (Poisson or Gaussian?) Integrate over all a,b to get marginalised P(r,S) Read off desired limits by  0 n P(r,S) =1-CL etc Done approximately for small errors (Cousins and Highland). Shows that limits pretty insensitive to  a,  b Numerically for general errors (RB: java applet on SLAC web page). Includes 3 priors (for a) that give slightly different results

35. Results from Small Numbers35 3.3-3.9 Extend Feldman Cousins Profile Likelihood: Use P(S)=P(n,S,a max,b max ) where a max,b max give maximum for this S,n Empirical Bayes And more… Results being compared as outcome from Banff workshop

36. 36 Summary Straight Frequentist approach is objective and clean but sometimes gives ‘crazy’ results Bayesian approach is valuable but has problems. Check for robustness under choice of prior Feldman-Cousins deserves more widespread adoption Lots of work still going on This will all be needed at the LHC