# Presentation: 227600

## CERN Higgs searches: CLs

**W. J. Murray**
**RAL**

## Talk overview

**Definition of CL****s**
**Application in Higgs search**
**What about Discovery?**
**Nuisance Parameters**
**Techniques for calculation**
**Conclusion**

## Requirements of a CL

**Initially seen from a ****frequentist**** perspective**
**Modified by ****Bayesian**** interpretation**
**Need to be acceptable to community So must satisfy ****BOTH**** schools**
**Nb Powerpoint thinks both frequentist and Bayesian are spelt wrong..**

## Definition of CLb and CLsb

**Frequentist Definition:**
**Background ensemble**
**Signal +Back ensemble**
- 2 hypotheses considered, and only two!
- Ordering automatically 1 sided

## Definition of CLs

**CL****s**** is a safer CL****sb**
**Used only to Exclude a signal**
**CL****sb**** was frequentist CL, CL****s**** is *****LARGER***** so ****conservative                           ****- Frequentist-safe**
**Asks `How much ****more**** unlikely from s than b?’ ****l****ike LR:                                        ****- Bayes-like**

## Definition of CLb’

**CL****b****’**** is a safer CL****b**
**Used only to Discover a signal**
**CL****b**** was frequentist CL, CL****b****’**** is *****SMALLER***** so ****conservative                           ****- Frequentist-safe**
**Asks `How much ****more**** unlikely from b than s?’ ****l****ike LR:                                             ****- Bayes-like**

## Typical PDF distribution:

- CLs and
- CLsb similar

- Low ll: Exclusion
- medium ll:    no conclusion
- high ll: Discovery

- Cls always increase by construction

## Clear PDF distribution:

- Clear PDF distribution:

- If separation was much larger we would not use statistics

- CLs and
- CLsb identical

- Treatment of results outside either remains a potential problem!

## Insensitive PDF distribution:

- CLs and CLsb distinctly different

## Useless PDF distribution:

- CLsb allows exclusion. CLs does not.

- This is the case where CLsb feels wrong

- Rev. Bayes!

## Clear Poisson Distribution:

- 3 events observed

- Signal of 10 excluded

## Typical Poisson Distribution:

- Signal of 4

- For 3 seen, CLs is always ~twice CLsb

## Useless Poisson Distribution:

- Nb:
- With CLb, 0 observed is always excluded at e-s

- How did we find this definition?

**It was an extension of the RPP `96:**
**This is the same as CL****s**** for Poisson**

## Why not Feldmann & Cousins?

**There are drawbacks to F+C:**
**Limits below e****-s  ****when 0 events seen**
**Needs more information than we have! (Some experiments treat each Higgs mass as a separate search, and return `independent’ results)**
**Limit can benefit from fluctuations elsewhere **
**It has advantages**
**Solves the look-elsewhere**
**Not clear whether automatic 2-sided limits are an advantage**

## Summary of CLs

**Gives overcoverage for classical limits**
**Outperforms the Bayesian integral with a flat prior in signal rate**
**Deontologically acceptable - i.e. does not exclude where no discrimination**
**Does not just tell us whether it is raining**

- (C) P. Janot

## Application in Higgs search

**How powerful are the techniques?**
**Higgs rate v mass **
**          **

## Observed Confidence Levels

- LR is same as R value proposed by d’Agostini

- CLsb and CLb converge for good separation

## Expected Confidence Levels

- What does expected mean?
**Mean**
- Has normally been used by us.
**Median**
- No dependence on metric
- Careful: Both are used here!

- Expected limits:
- CLs .3GeV below CLsb
- LR: 1GeV below CLsb

## Probability of Exclusion

- Define exclusion as CLs<0.05
- Probability of false exclusion should be 5% - but is less
- Significant overcoverage

- False exclusion rate is always 5% *of the true exclusion rate*

## What about Discovery?

**CL****b**** is the accepted indicator (CL****b****’ under study)**
**Require 5******** => 1-CL****b**** < 5*10****-7**
**No real allowance for flip-flopping**
**Can (*****will*****!) ****ALWAYS**** quote limit**
**flip-flop probability ****VERY**** small**

## Look-elsewhere effect

**Can discover at ****ANY**** mass - raises probability of fake discovery above 5*10****-7**** **
**Results are each mass are correlated **
***But:***** What mass range should be checked? **
**Last years limit to sensitivity limit?**
**Full range scanned?**** (But that is arbitrary!)**
**No ****RIGHT**** answer - We use a down-weighting factor, from MC experiments; ****4 in SM case****.**

## Nuisance Parameters

**See Slides from T. Junk **

## Techniques for Calculation

**Three different methods used:**
**Monte Carlo calculation. (A.Read) Flexible but slow. Tricks help.****	**
**Analytic folding (P.Bock)**
**event by event: Good for low event nos.**
**bin by bin: Good for low bin nos.**
**FFT approach: (S.Nielsen) Fastest for large problems**

## Monte Carlo calculation

**Used for current LEP limits**
**Very easy to add all sorts of complications by varying ensemble**
**Takes several days CPU for MSSM limits.**
**Use ****LR=(s+b)/b**** to enhance**** effective ****statistics**

## Conclusions:

**CL****s**** is well-tested practical solution.**
**Safe for Classical statistician**
**Bayes-like properties for a Bayesian**
**Removes a few hundred MeV w.r.t. to ****optimal**** Frequentist CL****sb**
**No**** Higgs ****found yet (m****H****>107.7GeV)**