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

2. Everything you wanted to know about limits
Summary
Prediction
confronts data & sees
small/zero signal
Frequentist
probability and
Confidence
Level language
Likelihood
Bayesian
Probability
(Health
Warning)
Zero events
Gaussian
ln L= -½
Extension to
several
parameters
Few events:
Confidence belt
The horrendous
case of large
backgrounds
Roger Barlow: YETI06
Everything you wanted to know about limits

3. Everything you wanted to know about limits
Model predictions
Input model and
parameters
Low energy
Lagrangian
Cuts designed to
bring out signal
Feynman Rules
for Feynman
diagrams
Cross Sections
and Branching
Ratios
Experiment duration,
luminosity,
Efficiency etc
Monte Carlo
programs
Number
of events
Data
Roger Barlow: YETI06
Everything you wanted to know about limits

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…”
Roger Barlow: YETI06
Everything you wanted to know about limits

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)
Roger Barlow: YETI06
Everything you wanted to know about limits

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.
Roger Barlow: YETI06
Everything you wanted to know about limits

7. Limitation of Frequentist Probability
Can’t say
“It will probably rain tomorrow.”
There is only one tomorrow. P is either 1 or 0
Have to say
“The statement ‘It will rain tomorrow.’ is probably true.”
Can then even quantify (meteorology).
Roger Barlow: YETI06
Everything you wanted to know about limits

8. Interpreting physics results: Mt =173±2 GeV/c2
Can’t say
‘Mt has a 68% probability of lying between 171 and 175 GeV/c2’
Have to say
‘The statement “Mt lies between 171 and 175 GeV/c2”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 “Mt lies between 171 and 175 GeV/c2” with 68%
Confidence. Or with 95% confidence. Or…
Roger Barlow: YETI06
Everything you wanted to know about limits

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
Roger Barlow: YETI06
Everything you wanted to know about limits

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, MT, MH, whatever
Interpret physics results using Bayes’ Theorem:
pposterior(a|data)  p(data|a) x pprior(a)
Roger Barlow: YETI06
Everything you wanted to know about limits

11. Everything you wanted to know about limits
Bayes at work
Zero events seen x =   
Posterior P()
P(0 events|)
(Likelihood)
Prior: uniform
3 P() d= 0.95
Same as Frequentist limit - Happy coincidence
Roger Barlow: YETI06
Everything you wanted to know about limits

12. Bayes at work again Is that uniform prior really credible? x =   
Prior: uniform in ln l
P(0 events|)
Posterior P()
Upper limit totally different!
3 P() d >> 0.95
Roger Barlow: YETI06
Everything you wanted to know about limits

13. Everything you wanted to know about limits
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’.)
Roger Barlow: YETI06
Everything you wanted to know about limits

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.
Roger Barlow: YETI06
Everything you wanted to know about limits

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   with 95% probability
is below x  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
Roger Barlow: YETI06
Everything you wanted to know about limits

16. Everything you wanted to know about limits
A Poisson measurement
You detect 5 events. Best value 5. But what about the errors?
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
Roger Barlow: YETI06
Everything you wanted to know about limits

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 
68%
16%
16%
 - n
Technique works for any CL, and single or double sided
Roger Barlow: YETI06
Everything you wanted to know about limits

18. Everything you wanted to know about limits
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
Roger Barlow: YETI06
Everything you wanted to know about limits

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 Or Or 10.9 ?
“We say at 68% confidence that the number of signal events lies between 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
Roger Barlow: YETI06
Everything you wanted to know about limits

20. Everything you wanted to know about limits
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=nexp(-+-b) (++b)r/r!
nexp(-b) br/r!
Known as “the old PDG formula” or “Helene’s formula” or “that heap of crap”
Roger Barlow: YETI06
Everything you wanted to know about limits

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
Roger Barlow: YETI06
Everything you wanted to know about limits

22. Everything you wanted to know about limits
Strategy 3: CLs
As used by LEP Higgs working group
Generalisation of Helene formula
Some quantity Q. Could be number of events, or something more clever
CLb=P(Q or less|b) CLs+b=P(Q or less|s+b)
CLs=CLs+b/CLb
Used as confidence level. Optimise strategy using it and quote results
Roger Barlow: YETI06
Everything you wanted to know about limits

23. Everything you wanted to know about limits
2(+) parameters
L(a,b)
Fix b, find 68% confidence range for a, using ln L=-½
Fix a, find 68% range for b
Combination (square) has 0.682=46% b a
ln L=-½ circle has 39% Confidence
Define regions through contours of log L – Confidence content given by 2ln L= 2 for which P(2 ,n)=CL
Caution! Cannot redefine a as b+c+d, claim 3 parameters and cut with P(2,3) instead of P(2 ,1)
Roger Barlow: YETI06
Everything you wanted to know about limits

24. Everything you wanted to know about limits
Summary
Prediction
confronts data & sees
small/zero signal
Frequentist
probability and
Confidence
Level language
Likelihood
Bayesian
Probability
(Health
Warning)
Zero events
Gaussian
ln L= -½
Extension to
several
parameters
Few events:
Confidence belt
The horrendous
case of large
backgrounds
Roger Barlow: YETI06
Everything you wanted to know about limits

25. Everything you wanted to know about limits
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.
Roger Barlow: YETI06
Everything you wanted to know about limits

26. Everything you wanted to know about limits
Further Reading
Workshop on Confidence Limits, CERN yellow report
Proc. Conf. Advanced Statistical Techniques in Particle Physics, Durham, IPPP02/39
Proc. PHYSTAT03 – SLAC-R-703
Proc PHYSTAT05, Oxford - forthcoming
Roger Barlow: YETI06
Everything you wanted to know about limits