A Bayesian Analysis of Parton Distribution Uncertainties Clare Quarman Atlas UK Physics meeting – UCL 15 th 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

How PDFs are calculated Initial parameterisation (low energy, Q 0 2 ) e.g. DGLAP evolution (to energy of data) where each Comparison with data Adjust parameters to give best fit

DGLAP evolution my LO evolution code using MRST initial distributions MRST 2001 LO

DGLAP evolution my LO evolution code using MRST initial distributions MRST 2001 LO

DGLAP evolution my LO evolution code using MRST initial distributions MRST 2001 LO

PDF Uncertainties: Current Status Majority frequentist: –MRST papers on both theory and expt errors Eur.Phys.J. C28 (2003) 455 [hep-ph/ ] [hep-ph/ ] –CTEQ uncertainties JHEP 0207 (2002) 012 [hep-ph ] Bayesian: –W. Giele & S. Keller Phys.Rev. D58 (1998) [hep-ph/ ] – expt, NLO [hep-ph/ ] – 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.

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing a coin What is the heads/tails bias? Taken from: Data Analysis: a Bayesian Tutorial, DS Sivia (OUP 1996)

simple Bayesian example Tossing 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

… How it will work Step 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 recall:

…How it will work… Width uncertainty in prediction of

… How it will work Step 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 recall:

Incompatible Data Sets Choice 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

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 Not naturally provided by a Bayesian analysis G(x) a very flexible function

Status Very 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