Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 1 Upper Limits and Discovery in Search for Exotic...

Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 1

Upper Limits and Discovery in Search for Exotic Physics

Jan Conrad

Royal Institute of Technology (KTH)

Stockholm


Outline

Discovery Confidence Intervals The problem of nuisance parameters

(“systematic uncertainties”)

Averaging Profiling

Analysis optimization

Summary


General approach to claiming discovery (hypothesis testing)

Assume an alleged physics process characterized by a signal parameter s (flux of WIMPS, Micro Blackholes .... etc.)

One can claim discovery of this process if the observed data is very unlikely to come from the null hypothesis , H0, being defined as non-existence of this process (s=0). ”Very unlikely” is hereby quantified as the ”signifcance” probability αsign, taken to be a small number (often 5 σ ~ 10-7).

Mathematically this is done by comparing the p-value with αsign and reject H0 if p –value < αsign

test statistics, T, could be for example χ2

Actually observed value of the test statistics


P-values and the Neyman Pearson lemma

Uniformly most powerful test statistic is the likelihood ratio :

For p-values, we need to know the null-distribution of T.

Therefore it comes handy that asymptotically:

Often it is simply assumed that the null-distribution is χ2 but be careful !

see e.g. J.C. , presented at NuFACT06, Irvine, USA, Aug. 2006 L. Demortier, presented at BIRS, Banff, Canada, July 2006


Type I, type II error and power

Type I error: Reject H0, though it is true.

Prob(Type I error) = α

Type II error: Accept H0, though it is false

Power: 1 - β = 1 – Prob(Type II error)

In words: given H1, what is the probability that we will reject H0 at given

significance α ? In other words: what is the probability that we detect H1 ?

In designing a test, you want correct Type I error rate (this controls the number of false detections) and as large power as possible .


Why 5 ?

… traditional: we have seen 3 significances disappear (….we also

have seen 5 signficances disappear on the other hand ….)

Principal reasoning (here done for the LHC):

LHC searches: 500 searches each of which has 100 resolution elements (mass, angle bins, etc.) 5 x 104 chances to find something.

One experiment: False positive rate at 5 (5 x 104) (3 x 10-7) = 0.015. OK !

Two experiments: Assume we want to produce < 100 unneccessary theory papers allowable false positive rate: 10. 2 (5 x 104) (1 x 10-4) = 10 3.7 required. Required other experiment verification:

(1 x 10-3)(10) = 0.01 3.1 required.

It seems that the same reasoning would lead to smaller required signficance probabilities for EP searches in NT.


Confidence Intervals (CI)

Instead of doing a hypothesis test, we might want to do a interval estimate on the parameter s with confidence level 100(1 – α) % (e.g. 90 %):

Bayesian:

Frequentist:

Invert by e.g. Neyman construction of confidence intervals (no time to explain)

- special case 1: n 2= upper limit - special case 2: two sided/one sided limits depending on observation Feldman & Cousins

Confidence intervals are often used for hypothesis testing.

G. Feldman & R. Cousins, Phys. Rev D57:3873-3889

See e.g. J.C. presented at NuFACT06, Irvine, USA, Aug. 2006

K. S. Cranmer, PhyStat 2005, Oxford, Sept. 2005


Nuisance parameters1)

Nuisance parameters are parameters which enter the data model, but which are not of prime interest (expected background, estimated signal/background efficiencies etc. pp., often called systematic uncertainties)

You don’t want to give CIs (or p-values) dependent on nuisance parameters need a way to get rid of them

1) Applies to both confidence intervals and nuisance parameters


How to ”get rid” of the nuisance parameters ?

There is a wealth of approaches to dealing with nuisance parameters. Two are particularly common:

Averaging (either the likelihood or the PDF):

Profiling (either the likelihood or the PDF):

... less common, but correct per construction: fully frequentist, see e.g:

Bayesian

G. Punzi, PHYSTAT 2005, Oxford, Sept. 2005


NT searches for EP: why things are bad ..... and good.

Bad Low statistics makes the use of asymptotic methods

doubtful systematic uncertainties are large.

Good: Many NT analyses are single channel searches with

relatively few nuisance parameters

rigorous methods are computationally feasible (even fully frequentist)


Coverage

A method is said to have coverage (1-α) if, in infinitely many repeated experiments the resulting CIs include (cover) the true value in a fraction (1-α) of all cases (irrespective of what the true value is).

Coverage is a necessary and sufficient condition for a valid CI calculation method

1 -α

s

1

0.9over-covering

under-covering


Averaging: hybrid Bayesian confidence intervals

J.C, O. Botner, A. Hallgren, C. de los Heros Phys. Rev D67:012002,2003 R. Cousins & V. Highland Nucl. Inst. Meth. A320:331-335,1992

Example PDF:

Perform Neyman-Construction with this new PDF (we will assume Feldman & Cousins in the remainder of this talk)

Treats nuisance parameters Bayesian, but performs a frequentist construction.

Integral is performed in true variables Bayesian


Coverage of hybrid method.

J.C & F. Tegenfeldt , PhyStat 05, Oxford, Sept. 2005, physics/0511055

(1-

α) M

C

true s true sF.Tegenfeldt & J.C. Nucl. Instr. Meth.A539:407-413, 2005

Use Log-normal if large uncertainties !!!!!


Commercial break: pole++

Bayesian treatment in FC ordering Neyman construction treats P(n|εs +b)

Consists of C++ classes: Pole calculate limits Coverage coverage studies Combine combine experiments

Nuisance parameters supports flat, log-normal and Gaussian uncertainties in efficiency and

background Correlations (multi-variate distributions and uncorrelated case)

Code and documentation available from: http://cern.ch/tegen/statistics.html

J.C & F. Tegenfeldt , Proceedings PhyStat 05, physics/0511055


Example: hybrid Bayesian in NTs

From Daan Huberts talk (this conference):

with systematicswithout systematics


Profiling: Profile Likelihood confidence intervals

Lower limit Upper Limit

2.706

meas n, meas. b MLE of b given s

MLE of b and s given observations

To extract limits:


From MINUIT manual

See F. James, MINUIT Reference Manual, CERN Library Long Write-up D506, p.5:

“The MINOS error for a given parameter is defined as the change in the value of the parameter that causes the F’ to increase by the amount UP, where F’ is the minimum w.r.t to all other free parameters”.

Confidence Interval

Profile Likelihood ΔΧ2 = 2.71 (90%),

ΔΧ2 = 1.07 (70 %)


Coverage of profile likelihood

Background: Poisson (unc ~ 20 % -- 40 %) , Efficiency: binomial (unc ~ 12%) Rolke

et alMinuit

W. Rolke, A. Lopez, J.C. Nucl. Inst.Meth A 551 (2005) 493-503

(1-

α) M

C

true s

Available as TRolke in ROOT !

Should be able to treat common NT cases


Profile likelihood goes LHC.

Basic idea: calculate 5 σ confidence interval and claim discovery if s = 0 is not included.

Straw-man model:

Typical: b = 100, т = 1 ( 10 % sys. Uncertainty on b)

K. S. Cranmer, PHYSTAT 2005, Oxford, Sept. 2005

Method Calculated coverage (nom. 5σ)

Critical number of events (bobs = 100)

Ignore ”systematics” 3.1 150

Bayesian treatment 4.2 178

Profile likelihood 5.0 185

Size of side band region

- 17 events!!

- 35 events!!


Analysis optimisation

Consider some cut-value t. Analysis is optimised defining a figure of merit (FOM). Very common:

Alternatively, optimize for most stringent upper limit. The corresponding figure of merit is the model rejection factor, MRF:

G. Hill & K. Rawlins, Astropart. Phys. 19:393-402,2003

Mean upper limit (only bg)


In case of systematics ?

Simplest generalizations one could think of:

In general, I do not think it makes a difference unless:

NO !

Yes !


Optimisation for discovery and upper limit at the same time ?

Fix significance (e.g αsign = 5 σ) and confidence level (e.g. 1-αCL = 99 %). Then define sensitivity region in s by :

The FOM can be defined to optimize this quantity (e.g simple counting experiment):

G. Punzi, PHYSTAT 2003, SLAC, Aug. 2003

Signal efficiency

Number of σ (here assumed αsign = 1 – αCL)


Conclusions/Final Remarks

Two methods to calculate CI and claim discovery in presence of ”systematic” uncertainties have been discussed.

The methods presented here are certainly suitable for searches for Exotic Physics with Neutrino Telescopes and code exists which works ”out of the box”

Remark: the ”simplicity” of the problem (single channel, small number of nuisance parameters) make even rigorous methods applicable

Remark 2: the LHC example shows that for large signficances (discovery) hybrid Bayesian might be problematic.

I discussed briefly the issue of sensitivity and analysis optimisation.


Backup Slides


B0s µ+µ-

CDF1 CDF 2

Eff uncertainty [%] 18.2 16.0

Eff uncertainty [%] 20.3 19.2

Corr. eff. Uncertainty [%]

15.5

95 % CI [10-7] 2.5 4.3

95 % combined limit [10-7]

1.7 (2.0 in Bernhard et. al.)

J.C & F. Tegenfeldt , Proceedings PhyStat 05, physics/0511055


Neyman construction

J. Neyman, Phil. Trans. Roy. Soc. London A, 333, (1937)

Exp 1Exp 2

Exp 3

One additional degree of freedom: ORDER in which you inlcude the n into the belt


Average coverage

s

Max/Min coverage

Ordering function: (Punzi, PhyStat05)

Can be any ordering in prime observable sub-space, in this case

Likelihood ratio (Feldman & Cousins)

~ FC Profile

s

Poisson signal, Gauss eff. Unc (10 %)

Projection method with appropriate ordering.


FC ordering: coverage

(1-

α) M

C

true s

Nominal coverage

Calculated by Pseudo-experiments


Some methods for p-value calculation

Conditioning Prior-predictive Posterior-predictive Plug-In Likelihood Ratio Confidence Interval Generalized frequentist


Some methods for confidence interval calculation (the Banff list)

Bayesian Feldman & Cousins with Bayesian treatment of nuisance

parameters (Hybrid Bayesian) Profile Likelihood Modified Likelihood Feldman & Cousins with Profile Likelihood Fully frequentist Empirical Bayes

Jan Conrad (KTH, Sthlm) EPNT 06 20 September 2006 1 Upper Limits and Discovery in Search for Exotic...

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