1. Confidence Intervals and Limits
Construction of confidence intervals
Frequentist and Bayesian approaches
“Unified” approach to CL
A look at the Higgs discovery
Elton S. Smith, Jefferson Lab

2. Peaks in reconstructed masses
Example of identifying positive particles in an interaction (inclusive)
g12C -> 1 charged+....
“misidentification” background present

3. Missing mass spectrum in gp -> p X
1. Physical background
2. Varying resolution
3. Different signal/background
Non-resonant
gp -> p 2p

4. Signal and background N = 1000 Clear signal
Estimate of signal size depends on assumptions about background.
Uncertainties depend on the (unknown) accuracy of background estimate.

5. Central Confidence Interval for Gaussian
Pick a value of the true parameter m. Compute the 90% CL range for that m.
Repeat at other m.
Area = 90%

6. Confidence levels and Gaussian tails

7. Signal and background N = 100 Dubious signal
Assumptions about background become critical.
Signal = Data – Background
Significance ~ Signal/Uncertainty
True signal ~ 100/√1865 = 2.3
Estimated signal ~ 174/42 = 4.1
Construct confidence interval at fixed values of true signal “m”

8. Setting limits

9. Upper limits
Same procedure as for central interval, but now for upper limits.
Area = 90%

10. Bayes Theorem

11. The law of total probability

12. The Monty Hall problem
P(1) is the probability car is behind door 1
Suppose you're on a game show, and you're given the choice of three doors:
Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?"
Is it to your advantage to switch your choice from door No. 1 to door No. 2?

13. The Monty Hall problem
Contestant picks door 1
Let B represent “the host opens door 3”.
With no prior knowledge P(B) = 0.5
Host opens door 3 when car is behind door 3
Contestant should always change from door 1 to door 2 and will double changes of winning!

14. Intuitive solution
The information that the host will not open the door with the car increases chances of winning.

15. Bayesian statistics: general philosophy

16. Flip-flopping physicist
To avoid unphysical region, use procedure based on data x:
x < 0, assume x=0
0 < x < 3, upper limits
x > 3, central confidence interval
True mean m (units of s)
-1.64
Unphysical
-1.28
Feldman Cousins Phys Rev D57 (1998) 3873
Measured mean x (units of s)

17. Probability density for Gaussian distribution
Wish to determine parameter m
Measure X, with known constant background b
Estimate for m is x
Function of x/s and m/s only

18. Likelihood ratio computed relative to probability with m=max(0,x)
Ordering principle
Likelihood ratio computed relative to probability with m=max(0,x)
Interval selected using the two conditions
and
Feldman Cousins Phys Rev D57 (1998) 3873

19. Likelihood Ratio: R(x) = P(x|m)/P(x/mbest)

20. Ordering Principle: Picking the 90% interval
Feldman Cousins Phys Rev D57 (1998) 3873

21. Feldman-Cousins confidence intervals
Using a likelihood method to smoothly change from a two-sided (central) confidence interval to a single-sided interval FC gives a “unified” prescription for determining confidence intervals.
Avoids the unphysical region
This description sets limits or confidence intervals in a consistent manner independent of the observed data.

22. Higgs Boson Start with a massless theory of electroweak interactions
Add spontaneous symmetry breaking in the gauge theory by introducing a scalar doublet field
Masses are generated for the W and Z bosons, and masses to the fermions through the Yukawa interaction.
-> Predicts the existence of the
Standard Model Higgs boson
ATLAS Phys Lett B 716 (2012) 1
CMS Phys Lett B 716 (2012) 30

23. Higgs boson search
Interactions are completely determined for a fixed value of the Higgs mass MH (just as all electromagnetic interactions are predicted for a fixed value of the electric charge).
Five Decay Modes
ATLAS Phys Lett B 716 (2012) 1
CMS Phys Lett B 716 (2012) 30

24. Expectations
The probability for a background fluctuation to be at least as large as the observed maximum excess is termed the local p-value

25. H->ZZ

26. H -> ZZ events
Naïve expectation ~ (9-3.8)/sqrt(3.8) ~ 2.7s

27. Significance of CMS H->ZZ?
p-value =

28. Published p-value for H->ZZ
Higher significance using distribution shape
Full significance includes all 5 decay modes

29. Summary Construction of confidence intervals.
Frequentist (classical) and Bayesian approaches
Description of the “unified” approach to constructing confidence intervals.
Discussion of the CMS data supporting the discovery of the Higgs boson