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PDF Errors with Normalization Uncertainties
Hessian vs Monte Carlo The D’Agostini Bias Bias free Fitting NNPDF: RDB, Luigi Del Debbio, Stefano Forte, Alberto Guffanti, Jose Latorre, Juan Rojo, Maria Ubiali (Barcelona, Edinburgh, Freiburg, Milan) PDF4LHC CERN August 2009
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(NN)PDFs for LHC No bias due to functional form
To fully exploit LHC data, we need: Precise reliable faithful PDFs No theoretical bias (beyond NLO pQCD, etc.) No bias due to functional form No bias due to improper statistical procedure Genuine statistical confidence level Full inclusion of correlations in exp systematics Full inclusion of normalization uncertainties No rescaling of experimental errors Uniform treatment of uncertainties
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(NN)PDFs for LHC Zero Tolerance! No bias due to functional form
To fully exploit LHC data, we need: Precise reliable faithful PDFs No theoretical bias (beyond NLO pQCD, etc.) No bias due to functional form No bias due to improper statistical procedure Genuine statistical confidence level Full inclusion of correlations in exp systematics Full inclusion of normalization uncertainties No rescaling of experimental errors Uniform treatment of uncertainties Zero Tolerance!
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Catalogue of (average) Errors
Process Expt Set Ndata Stat Syst Norm Fixed Target SLACp 211 2.7% 2.2% BCDMSp 351 3.2% 2.0% NMCp 288 3.7% 2.3% HERA Z97NC 160 6.2% 3.1% 2.1% Z02NC 92 12.7% 1.8% Z03NC 90 7.7% 3.3% Z06NC 3.8% 2.6% H197NC 130 12.5% 1.5% H199NC 126 14.9% 2.8% H100NC 147 9.4% Nu-DIS CHORUSnu 607 4.2% 6.4% NTVnuDMN 45 17.2% 1.0% Drell-Yan DYE605 119 9.9% DYE886p 184 19.9% 5.0% 6.5% Incl Jets CDFR2KT 76 4.5% 21.1% 5.7% D0R2CON 110 3.6% 6.1% S M A L 1.0 1.2 B I G 2.0
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Hessian Method cov0
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Monte Carlo Method cov0
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(for additive Gaussian uncertainties)
Hessian ´ Monte Carlo (for additive Gaussian uncertainties)
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Norm. in Monte Carlo: 1 expt
cov0
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Norm. in Monte Carlo: 2 expt
cov0
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Including Norm. Errors in 2
(Problems)
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Norm. in covariance: 1 expt
m-cov DISASTER! D’Agostini 1994
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Dashed line: data below 36 GeV
m-cov R(e+e-) CELLO 1987 Dashed line: data below 36 GeV Solid line: all data D’Agostini 1994
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Norm. in covariance: 2 expt
m-cov
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Norm. in covariance: 2 expt
m-cov D’Agostini 1994
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The Penalty Trick: 1 expt
n-cov
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The Penalty Trick: 2 expt
n-cov
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The Penalty Trick: 2 expt
n-cov
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Towards a Perfect 2 The Solution
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Unbiased covariance : 1 expt
t0-cov
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Unbiased covariance : 2 expt
t0-cov
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Unbiased covariance : 2 expt
t0-cov
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What is t0? t0-cov
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Nr2) 1-2r2 n-cov t0-cov
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Nr2) 1-2r2 n-cov t0-cov
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Nr2) 1-2r2 n-cov t0-cov
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Nr2) 1-2r2 n-cov t0-cov
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Important for balancing DIS and hadronic data
Summary D’Agostini bias: take care! Hessian Penalty Trick: bias » 1% t0-cov : unbiased fits for Monte Carlo (or Hessian) Important for balancing DIS and hadronic data
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Nr2) 1-2r2 n-cov t-cov 1+r2 t0-cov
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Nr2) 1-2r2 n-cov t-cov 1+r2 t0-cov
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Ns2) 1-2r2 n-cov t-cov 1+r2 t0-cov
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Summary E[t] Bias s1=s2 1=2 Decoupling s2m2 À 2 1 expt N expts
cov0 1 m-cov 1/(1+Nr2) 1-2r2 n-cov t-cov 1+r2 t0-cov
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