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

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

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

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.

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%

Confidence levels and Gaussian tails

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”

Setting limits

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

Bayes Theorem

The law of total probability

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?

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!

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

Bayesian statistics: general philosophy

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)

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

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

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

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

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.

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

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

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

H->ZZ

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

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

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

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