1. 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

2. (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

3. (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!

4. 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

5. Hessian Method
cov0

6. Monte Carlo Method
cov0

7. (for additive Gaussian uncertainties)
Hessian ´ Monte Carlo
(for additive Gaussian uncertainties)

8. Norm. in Monte Carlo: 1 expt
cov0

9. Norm. in Monte Carlo: 2 expt
cov0

10. Including Norm. Errors in 2
(Problems)

11. Norm. in covariance: 1 expt
m-cov
DISASTER!
D’Agostini 1994

12. 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

13. Norm. in covariance: 2 expt
m-cov

14. Norm. in covariance: 2 expt
m-cov
D’Agostini 1994

15. The Penalty Trick: 1 expt
n-cov

16. The Penalty Trick: 2 expt
n-cov

17. The Penalty Trick: 2 expt
n-cov

18. Towards a Perfect 2
The Solution

19. Unbiased covariance : 1 expt
t0-cov

20. Unbiased covariance : 2 expt
t0-cov

21. Unbiased covariance : 2 expt
t0-cov

22. What is t0?
t0-cov

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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