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PHYSTAT05 Highlights: Statistical Problems in Particle Physics, Astrophysics and Cosmology

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PHYSTAT05 Highlights:Statistical Problems in Particle Physics, Astrophysics and Cosmology

Phystat05 Highlights

University College London

03/11/2006

Müge Karagöz Ünel

Oxford University

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- Conference Information and History
- Introduction to statistics
- Selection of hot topics
- Available tools
- Astrophysics and cosmology
- Conclusions

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Poster

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7 Invited talks by Statisticians

9 Invited talks by Physicists

38 Contributed talks

8 Posters

Panel Discussion

3 Conference Summaries

90 participants

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David CoxKeynote Address: Bayesian, Frequentists & Physicists

Steffen LauritzenGoodness of Fit

Jerry FriedmanMachine Learning

Susan Holmes Visualisation

Peter CliffordTime Series

Mike TitteringtonDeconvolution

Nancy ReidConference Summary (Statistics)

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Bob CousinsNuisance Parameters for Limits

Kyle CranmerLHC discovery

Alex Szalay Astrophysics + Terabytes

Jean-Luc Starck Multiscale geometry

Jim LinnemannStatistical Software for Particle Physics

Bob NicholStatistical Software for Astrophysics

Stephen JohnsonHistorical Transits of Venus

Andrew JaffeConference Summary (Astrophysics)

Gary FeldmanConference Summary (Particles)

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Bayes/Frequentist5 talks

Goodness of Fit5

Likelihood/parameter estimation6

Nuisance parameters/limits/discovery10

Machine learning7

Software8

Visualisation1

Astrophysics5

Time series1

Deconvolution3

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An experiment goes through following stages:

- Prepare conditions for taking data for a particle X ( if theory driven)
- Record events that might be X and reconstruct the measurables
- Select events that could have X by applying criteria (cuts)
- Generate histograms of variables and ask the questions:
Is there any evidence for new things or is the null hypothesis unrefuted? If there is evidence, what are the estimates for parameters of X?(Confrontation of theory with experiment or v.v.)

- The answers can come via your favorite statistical technique (depends on how you ask the question)

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- Homo apriorius establishes probability of an hypothesis, no matter what data tell.
- Homo pragamiticus establishes that it is interested by the data only.
- Homo frequentistus measures probability of the data given the hypothesis.
- Homo sapiens measures probability of the data and of the hypothesis.
- Homo bayesianis measures probability of the hypothesis, given the data.

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We need to make a statement about Parameters, given Data

Bayes 1763 Frequentism 1937

Both analyse data (x) statement about parameters ( )

Both use Prob (x; ), e.g. Prob ( ) = 90%

but very different interpretation

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Bayesian :Probability (parameter, given data)

Frequentist :Probability (data, given parameter)

“Bayesians address the question everyone is interested in, by using assumptions no-one believes”

“Frequentists use impeccable logic to deal with an issue of no interest to anyone”

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LauritzenInvited talk - GoF

Yabsley GoF and sparse multi-D data

IanniGoF and sparse multi-D data

RajaGoF and L

Gagunashvili2and weighting

PiaSoftware Toolkit for Data Analysis

Block Rejecting outliers

Bruckman Alignment

BlobelTracking

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- We would like to: know if a given distribution is of a specified type, test the validation of a postulated model,..
- A few GoF tests are widely used in practice:
- 2 test: most widely used application is 1 or 2D fits to data
- G2 (the likelihood ratio statistics) test: the general version of 2 test (Lauritzen’s personal choice)
- Kolmogorov-Smirnov test: a robust but prone to mislead test, can be used to confirm, say, two distributions (histograms) are the same by calculating the p-value for the difference hypothesis.
- Other new methods, like Aslan&Zech’s energy test, exist…

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track

Intrinsic measurement error + MCS

hit

residual

Key relation!

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Direct Least-Squares solution to the Silicon Tracker alignment problem

The method consists of minimizing the giant 2resulting from a simultaneous fit of all particle trajectories and alignment parameters:

Let us consequently use the linear expansion (we assume all second order derivatives are negligible). The track fit is solved by:

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while the alignment parameters are given by:

Systems large: inherent

Computational challenges

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Equivalent to Millepede approach from V. Blobel

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Cousins Limits and Nuisance Params

Reid Respondent

Punzi Frequentist multi-dimensional ordering rule

Tegenfeldt Feldman-Cousins + Cousins-Highland

Rolke Limits

Heinrich Bayes + limits

Bityukov Poisson situations

Hill Limits v Discovery (see Punzi @ PHYSTAT2003)

Cranmer LHC discovery and nuisance parameters

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Note:Systematic errors (HEP) <-> nuisance params (statistician)

An example:

we need to know these, probably from other measurements (and/or theory)

Uncertainties error in

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Physics parameter

Observed

for statistical errors

Some are arguably statistical errors

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- Nuisance parameters are parameters with unknown true values. They may be:
- statistical, such as number of background events in a sideband used for estimating the background under a peak.
- systematic, such as the shape of the background under the peak, or the error caused by the uncertainty of the hadronic fragmentation model in the Monte Carlo.
- Most experiments have a large number of systematic uncertainties.
- If the experimenter is blind to these uncertainties, they become a bigger nuisance!

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- LHC will collide 40 million times/sec and collect petabytes of data. pp collisions at 14 TeV will generate events much more complicated than LEP, TeVatron.
- Kyle Cranmer has pointed out that systematic issues will be even more important at the LHC.
- If the statistical error is O(1) and systematic error is O(0.1), it does not much matter how you treat it.
- However, at the LHC, we may have processes with 100 background events and 10% systematic errors, this is not negligible.
- Even more critical, we want 5s for a discovery level.

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- LHC searches: 500 searches each of which has 100 resolution elements (mass, angle bins, ...) = 5 x 104 chances to find something.
- One experiment: False positive rate at 5 s(5 x 104) (3 x 10-7) = 0.015. OK.
- Two experiments:
- Assume allowable false positive rate: 10.
- 2 (5 x 104) (1 x 10-4) = 10 3.7 s required.
- Required other experiment verification, assume rate 0.01: (1 x 10-3)(10) = 0.01 3.1 s required.

- Caveats: Is the significance real? Are there common systematic errors?

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- Various techniques discussed during conference. Most concerns were summarized by Feldman.
- Bayesian: good method but Heinrich showed that flat priors in multi-D may lead to undesirable results (undercoverage).
- Frequentist-Bayesian hybrids: Bayesian for priors and frequentist to extract range. Cranmer considered this for LHC (which was also used at Higgs searches).
- Profile likelihood: shown by Punzi to have issues when distribution is Poisson-like.
- Full Neyman reconstruction: Cranmer and Punzi attempted this, but is not feasible for large number of nuisance parameters.

- Banff workhsop of this summer was found useful in comparing various methods. The real suggestions for LHC will likely come from 2007 workshop on LHC issues.

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- The problem: Given a measurement of an event X find F(X) which returns 1 if the event is signal (s) and 0 if the event is background (b) to optimize a figure of merit, say, s/√b for discovery and s/ √(s+b) for established signal.
- Theoretical solution: Use MC to calculate the likelihood ratio Ls(X)/Lb(X) and derive F(X) from it. Unfortunately, this does not work as in a high-dimension space, even the largest data set is sparse. (Feldman)
- In recent years, physicists have turned to machine learning: give the computer samples of s and b events and let the computer figure out what F(X) is.

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FriedmanMachine learning

ProsperRespondent

Narsky Bagging

RoeBoosting (Miniboone)

Gray Bayes optimal classification

Bhat Bayesian networks

Sarda Signal enhancement

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Various methods exist to classify, train and test events.

- Artificial neural networks (ANN): currently the most widely used (examples from Prosper, …)
- Decision trees: differentiating variable is used to separate sample into branches until a leaf with a preset number of signal and background events are found.
- Trees with rules: combining a series of trees to increase single decision tree power (Friedman)
- Bagging (Bootstrap AGGregatING) trees: build a collection of trees by selecting a sample of the training data (Narsky)
- Boosted trees: a robust method that gives misclassified events in one tree a higher weight in the generation of a new tree
Comparisons of significance were performed, but not all of were controlled experiments, so conclusions may be deceptive until further tests..

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Boosting the tree

Decision tree

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- An nice example from MiniBoone
- Create M many trees and take the final score for signal and background as weighted sum of individual trees

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Giovanni Punzi @ PHYSTAT2003

“Comments on L fits with variable resolution”

Separate two close signals (A and B) , when resolution σvaries event by event, and is different for 2 signals

e.g. M, Different numbers of tracks different σM

Avoiding Punzi bias

- Include p(σ|A) and p(σ|B) in fitOR
- Fit each range of σi separately, and add (NA)i (NA)total, and similarly for B
Beware of event-by-event variables and construct likelihoods accordingly

(Talk by Catastini)

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Potential problem: Experimenters’ bias

Original suggestion? Luis Alvarez

Methods of blinding:

- Keep signal region box closed
- Add random numbers to data
- Keep Monte Carlo parameters blind
- Use part of data to define procedure
A number of analyses in experiments doing blind searches

Don’t modify result after unblinding, in general..

Question: Will LHC experiments choose to be blind? In which analysis?

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Cosmologists

“Astronomers”

Particle Physicists

Bayesians

Frequentists

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‘“There is only one universe” and some experiments can never be rerun’ – A. Jaffe (concluding talk)

Astro+cosmo tend to be more Bayesian, by nature.

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- Virtual Observatories: all astro data available from desktop
- Data volume growth doubling every year, most data are on the web (Szalay)
- Bad: computing & storage issues
- Good (?): Systematic errors more significant statistical errors

- Nichol discussed using grid techniques.

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- Flat priors have been used commonly, but are dangerous (Cox, Le Diberder, Cousins): would be the best quantity to use or is it h2 ?
- Issues with non-gaussian distribution of noise taken into account in the spectrum: a few methods discussed by Starck, Digel, ..
- Blind analyses are rare (not so good at a priori modeling!)
- Lots of good software in astrophysics and repositories more advanced than PP.
- Jaffe’s talk has a a nice usage of CMB as a case study for statistical methods in astrophysics, starting from 1st principles of Bayesian.

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LinnemannSoftware for Particles

NicholSoftware for Astro (and Grid)

Le DiberdersPlot

Paterno R

Kreschuk ROOT

Verkerke RooFit

Pia Goodness of Fit

Buckley CEDAR

Narsky StatPatternRecognition

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- A number of good software has become more and more available (good news for LHC!)
- PP and astro use somehow different softwares (IDL, IRAF by astro, for ex.)
- 2004 Phystat workshop at MSU on statistical software (mainly on R & ROOT) by Linnemann
- Statatisticians have a repository of standard source codes (StatLib): http://lib.stat.cmu.edu/
- One good output of the conference was a Recommendation of Statistical Software Repository at FNAL
- Linnemann has a web page of collections: http://www.pa.msu.edu/people/linnemann/stat_resources.html

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- Documentation about statistics and a repository: http://www-cdf.fnal.gov/physics/statistics/statistics_home.html

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- Very useful physicists/statisticians interaction
- e.g. Confidence intervals with nuisance parameters,
- Multivariate techniques, etc..
- Lots of things learnt from
- ourselves (by having to present own stuff!)
- each other (various different approaches..)
- statisticians (update on techniques..)

- A step towards common tools/Software repositories: http://www.phystat.org(Linnemann)
- Programme, transparencies, papers, etc:http://www.physics.ox.ac.uk/phystat05(with useful links such as recommended readings)
- Proceedings published by Imperial College Press (Spring ’06)

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- A few workshops/schools took place since October, 2005
- e.g. Manchester (Nov 2005), SAMSI Duke (April 2006), Banff (July 2006), Spanish Summer School (July 2006)
- No PHYSTAT Conference in summer 2007
- ATLAS Workshop on Statistical Methods, 18-19 Jan 2007
- PHYSTAT Workshop at CERN, 27-29 June 2007 on
- “Statistical issues for LHC Physics analyses”.
- (Both workshops will likely aim at discovery significance. Please attend!)
- Suggestions/enquiries to: l.lyons@physics.ox.ac.uk

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- LHC will take data soon. We do not wish to say
- rather say

“The experiment was inconclusive, so we had to use statistics”

(inside cover of “the Good Book” by L. Lyons)

We used statistics, and so we are sure that we’ve discovered X

(well… with some confidence level!)

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- Tried to give you a collage of PHYSTAT05 topics.
- My deepest thanks to Louis for giving me the chance & introducing me to the PHYSTAT experience!
- Apologies to those talks I have not been able to cover…
- Thank you for the invitation!

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- Bayes
- Frequentist
- Cousins-Highland
- Higgs Saga at CERN

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Bayesian

Bayes’ Theorem

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posterior

likelihood

prior

Problems: P(param) True or False

“Degree of belief”

Prior What functional form?

Flat? Which variable?

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Neyman Construction

µ

x

x0

µ = Theoretical parameter

x = ObservationNO PRIOR

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at 90% confidence

Frequentist

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Bayesian

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Method: Mixed Frequentist - Bayesian

Full frequentist method hard to apply in several dimensions

Bayesian for nuisance parameters and

Frequentist to extract range

Philosophical/aesthetic problems?

Highland and Cousins

NIM A320 (1992) 331

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P (Data;Theory) P (Theory;Data)

Is data consistent with Standard Model?

or with Standard Model + Higgs?

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End of Sept 2000: Data not very consistent with S.M.

Prob (Data ; S.M.) < 1% valid frequentist statement

Turned by the press into: Prob (S.M. ; Data) < 1%

and therefore Prob (Higgs ; Data) > 99%

i.e. “It is almost certain that the Higgs has been seen”

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