Transcript ppt
A Bayesian Analysis of Parton 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 ( pp X ) ~ f i f j (ij X ) i, j How PDFs are calculated • Initial parameterisation (low energy, Q02) e.g. xg( x, Q0 ) axb (1 x)c (1 d x ex) 2 • DGLAP evolution (to energy of data) x 1 qi ( x, t ) S (t ) d Pqi q j ( , S (t )) t t g ( x, t ) 2 x Pgq j ( x , S (t )) Pqi g ( x , S (t )) q j ( , t ) x Pgg ( , S (t )) g ( , t ) where each Pab ( z, S ) Pab0 ( z ) 2 Pab1 ( z ) S • Comparison with data • Adjust parameters to give best fit DGLAP evolution Q0 1GeV 2 xf ( x, Q 2 ) 2 x my LO evolution code using MRST initial distributions MRST 2001 LO DGLAP evolution xf ( x, Q 2 ) Q 2 2 GeV 2 x my LO evolution code using MRST initial distributions MRST 2001 LO DGLAP evolution xf ( x, Q 2 ) Q 2 100 GeV 2 x 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/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 deals with What is it? outcome of a repeatable experiment Bayes theorem: parameters : data : meas prior beliefs quantifies degree of belief experiment posterior beliefs posterior likelihood prior p( | meas ) L( meas | ) ( ) • 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… parameters : (a, b,, S ) Step 1 data : meas • 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 • Calculate posterior recall: parameters : data : meas best fit parameters posterior likelihood prior p( | meas ) L( meas | ) ( ) Step 4 • Look at effect e.g. on W production cross section – generate many pdfs according to posterior distribution p( | meas ) – calculate W for each point histogram Step 5 • Vary priors and observe effect on results …How it will work… Width uncertainty in prediction of W W … How it will work Step 3 • Maximise likelihood • Calculate posterior recall: parameters : data : meas best fit parameters posterior likelihood prior p( | meas ) L( meas | ) ( ) Step 4 • Look at effect e.g. on W production cross section – generate many pdfs according to posterior distribution p( | meas ) – calculate W for each point histogram Step 5 • Vary priors and observe effect on results 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 replace s in likelihood – 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 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 Pab ( x, S ) • fit these parameters – posterior distribution should give indication of the size of the next order terms Goodness of fit ( Not naturally provided by a Bayesian analysis • how satisfactory are the initial distributions? • generalise by adding an extra term xg( x, Q0 ) axb (1 x)c (1 d x ex) G( x) 2 • put a prior on that it has a small value • posterior for should indicate goodness of fit ) 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