a bayesian analysis of parton distribution uncertainties clare quarman atlas uk physics meeting –...
TRANSCRIPT
A Bayesian Analysis ofParton Distribution
Uncertainties
Clare Quarman
Atlas UK Physics meeting – UCL 15th Dec 2003
Parton Distribution Functions(PDFs)
Tell us about
• the quark and gluon content of protons• how a proton’s momentum is distributed
between its constituents
Especially important now…
• hadron colliders – Tevatron and LHC– events caused by parton interactions– cross-sections depend on PDFs )(~)(
,
XijffXpp jji
i
How PDFs are calculated
• Initial parameterisation (low energy, Q02)
e.g.
• DGLAP evolution (to energy of data)
where each
• Comparison with data
• Adjust parameters to give best fit
),(
),(
))(,())(,(
))(,())(,(
2
)(
),(
),( 1
tg
tq
tPtP
tPtPdt
txg
txq
tt j
Sx
ggSx
gq
Sx
gqSx
x
Si
j
iji
)1()1(),( 20 exxdxaxQxxg cb
)()(),( 1
2
0 zPzPzP ababSabS
DGLAP evolution
my LO evolution code
using MRST initial distributionsMRST 2001 LO
220 GeV1Q
),
(2
Qx
xf
x
DGLAP evolution
my LO evolution code
using MRST initial distributionsMRST 2001 LO
22 GeV2Q
),
(2
Qx
xf
x
DGLAP evolution
my LO evolution code
using MRST initial distributionsMRST 2001 LO
22 GeV100Q
),
(2
Qx
xf
x
PDF Uncertainties: Current Status
Majority frequentist:– MRST papers on both theory and expt errors
• Eur.Phys.J. C28 (2003) 455 [hep-ph/0211080]
• [hep-ph/0308087]
– CTEQ uncertainties• JHEP 0207 (2002) 012 [hep-ph0201195]
Bayesian:– W. Giele & S. Keller
• Phys.Rev. D58 (1998) 094023 [hep-ph/9803393] – expt,
NLO• [hep-ph/0104052] – expt, theory, NLO
Frequentist Stats Bayesian Statistics– subjective probability
What is it?
Bayes theorem:
quantifies
degree of belief
deals with
outcome of a repeatable experiment
prior beliefs
experiment posterior beliefs
• Bayesian provides a framework for dealing with theoretical errors (unlike frequentist statistics)
• Theoretical errors dominate PDF uncertainties.
)()|()|(
measmeas Lp
priorlikelihoodposterior
measdata
parameters
:
:
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
simple Bayesian exampleTossing a coin
What is the heads/tails bias?
Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)
How it will work…
Step 1• identify priors
– use constraints– quantify more vague info– combine in a distribution of all parameters,
Step 2 - meanwhile…• predict deep inelastic scattering (DIS) cross section from
PDF (evolution: my LO code, QCDNUM NLO )
• calculate a likelihood function from DIS prediction and corresponding DIS data
meas
S
data
baparameters
:
),,,(:
)(
… How it will workStep 3• Maximise likelihood best fit parameters• Calculate posterior
Step 4• Look at effect e.g. on W production cross section
– generate many pdfs according to posterior distribution
– calculate for each point histogram
Step 5• Vary priors and observe effect on results
)()|()|(
measmeas Lp
priorlikelihoodposterior
measdata
parameters
:
:recall:
)|( measp
W
…How it will work…
W
Width
uncertainty in prediction of
W
… How it will workStep 3• Maximise likelihood best fit parameters• Calculate posterior
Step 4• Look at effect e.g. on W production cross section
– generate many pdfs according to posterior distribution
– calculate for each point histogram
Step 5• Vary priors and observe effect on results
)()|()|(
measmeas Lp
priorlikelihoodposterior
measdata
parameters
:
:recall:
)|( measp
W
Incompatible Data SetsChoice of data• influences the resulting best fit pdfs• some data sets seem to be incompatible• if one set is throwing the fit, when do you exclude it?• renormalisation scale errors
Our solution• assign
– a factor s that the uncertainty is underestimated by– a probability q of this happening
• put suitable priors on s and q• bayesian fit s and q along with all the other parameters
replace in likelihood s
Example problem: data with outlier
‘Good Data’
(Gausian distributed simulated data)
Least Squares Fit
Example problem: data with outlier
‘Bad Data’
one outlying point throws the fit
infact the mean has changed by more than the reported
error
Least Squares Fit
Example problem: data with outlier
‘Bad Data’
reported uncertainty is increased but the
mean is less affected
‘Goof factor’ fitted
Higher order terms• insert extra parameters representing the next unknown
order terms in splitting functions• fit these parameters – posterior distribution should give
indication of the size of the next order terms
Goodness of fit ( )
• how satisfactory are the initial distributions?• generalise by adding an extra term
• put a prior on that it has a small value• posterior for should indicate goodness of fit
)G()1()1(),( 20 xexxdxaxQxxg cb
),( Sab xP
Not naturally provided by a Bayesian analysis
G(x) a very flexible function
StatusVery much in the early stages, but so far..
Own LO DGLAP evolution program working
Very fruitful meeting, Durham Sept 2003• James Stirling (MRST partons)• Michael Goldstein (Bayesian statistician)
Most recently working on…C++ wrapping QCDNUM
integrating QCDNUM and my evolution code into next layer of the program which will allow comparison to data
Ultimately aim to make the whole program available to all - not just the parton sets