1. Incorporating systematic uncertainties into upper limits
Elton S. Smith, Jefferson Lab
Review of Feldman Cousins unified method for constructing confidence limits
Integrating over background level and detection efficiency
Summary and conclusions

2. Central Confidence Interval

3. Upper limits

4. Probability density Wish to determine parameter m
Measure X, with known constant background b
Estimate for m is x
Function of x/s and m/s only

5. Ordering principle
Likelihood ratio relative to probability with m=max(0,x)

6. Likelihood Ratio: R(x) = P(x|m)/P(x/mbest)
se=0

7. Ordering Principle: Picking the 90% interval
Feldman Cousins Phys Rev D57 (1998) 3873

8. Probability density, uncertainty in background
Background b, estimated at mb with uncertainty smb
Integrate over “true” background b
Function of x/s and m/s only

9. Probability density, scale uncertainty
Ŝ is the measured detection efficiency
S is the “true” detection efficiency
Define se = sS/Ŝ, the relative scale uncertainty

10. Probability distribution

11. Ordering principle
Likelihood ratio relative to probability with m=max(0,x)

12. 90% CL including scale uncertainties

13. Example 13

14. Correct Coverage se
90% CL

15. Dependence on se
Rolke NIMA 551 (2005) 493

16. Scaling from nominal FC limits
Simple procedure for scaling FC limits to include systematics

17. Summary and Conclusions
We have investigated how uncertainties in the estimation of background and detection efficiency affect the 90% confidence intervals in the unified approach of Feldman and Cousins.
Assumption: Gaussian statistics
Systematic uncertainties included using Bayesian approach
Confidence intervals have reasonable limiting behavior
depend quadratically on se
can be obtained by scaling the nominal FC intervals
have correct coverage
CLAS-NOTE and CLAS-NOTE