1. The asymmetric uncertainties on data points in Roofit
Few slides on ‘easy/trivial’ topic that will hopefully leave you confused
di-muon invariant mass [MeV/c2]
Candidates (50 MeV/c2)
CERN seminar Flavio Archilli
Ivo van Vulpen (Nikhef/UvA, ATLAS)

2. How are our asymmetric uncertainties on data points defined ?
Not ± √n 4
Candidates (50 MeV/c2)
di-muon invariant mass [MeV/c2]
2/20

3. statistics
3/20

4. You are responsible for how you summarize your measurement
Solving statistics problems in general
Discussions (in large experiments):
- strong opinions, very outspoken ‘experts’
- use RooSomethingFancy: made by experts & debugged
- let’s do what we did last time, let’s be conservative - …
You are responsible for how you summarize your measurement
The tools you use have assumptions, biases, pitfalls, … , so you know best how others (and you) should interpret your measurement
4/20

5. Six reasonable options
Discuss in your group which one you prefer
How should we represent this measurement ? 4
Candidates (50 MeV/c2)
di-muon invariant mass [MeV/c2]
Basis for this talk:
5/20

6. Option 1: assign NO uncertainty
± 0.00
The number of observed events is what it is: 4
The uncertainty is in the interpretation step, i.e. on the model-parameters that you extract from it
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7. Comfortable territory: Poisson distribution
Probability to observe n events when λ are expected
Binomial with n∞, p0 en np=λ
Number of observed events
#observed
λ hypothesis
P(Nobs|λ=4.9)
λ=4.90 4
7/20

8. Properties of the Poisson distribution
the famous √n
properties
λ=1.00
(1) Mean:
(2) Variance:
(3) Most likely: first integer ≤ λ
λ=4.90
λ=5.00
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9. Option 2: the famous sqrt(n)
+2.00
-2.00
Poisson variance for λ equal to measured number of events
… but Poisson distribution is asymetric:
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10. Just treat it like a normal measurement
What you have:
1) construct the Likelihood λ as free parameter
2) Find value of λ that maximizes the Likelihood
3) Determine error interval:
Δ(-2Log(Lik.)) = +1
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11. Likelihood
Likelihood Poisson: probability to observe n events when λ are expected
Likelihood
P(Nobs=4|λ)
#observed
λ hypothesis
λ (hypothesis)
fixed
scan
11/20

12. Option 3: Likelihood 12/20 P(Nobs=4|λ) Log(Likelihood)
+2.35
P(Nobs=4|λ)
-1.68
λ (hypothesis)
-2Log(L)
Log(Likelihood)
-1.68
+2.35
-2Log(P(Nobs=4|λ))
ΔL=+1
12/20
2.32
4.00
6.35

13. Bayesian: statement on value of λ
What you have:
What you want:
Likelihood
pdf for λ
Probability to observe Nobs events … given a specific hypothesis (λ)
What can we say about theory (λ) … given # of observed events (4)
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14. Bayesian: statement on value of λ
Choice of prior P(λ): (basis for religious wars)
Here: assume all values for λ equally likely
Likelihood
Probability density function
λ (hypothesis)
λ (hypothesis)
Posterior PDF for λ
 Integrate to get confidence interval

15. Option 4 & 5: representing the Bayesian posterior
Bayes central
+3.16
68%
16%
16%
-1.16
Bayes equal probability
+2.40
68%
8.3%
23.5%
-1.71
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16. Option 6: Frequentist approach
Find values of λ that are on border of being compatible with observed #events
λ=7.16
If λ > 7.16 then probability to observe 4 events (or less) <16%
16%
Note: also uses ‘data you didn’t observe’, i.e. a bit like definition of significance
smallest λ (>n) for which
P(n≤nobs|λ) ≤ 0.159
+3.16
λ=2.09
-1.91
largest λ (<n) for which
P(n≥nobs|λ) ≤ 0.159
16%
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17. The six options 8 7
-1.91
+3.16 6
+2.00
+2.35
+3.16
+2.40 5 4 3
-2.00
-1.68
-1.16
-1.71 2 1
Nothing
Poisson
Likelihood
Bayesian central
Bayesian equal prob
Frequentist
Δλ:
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18. The six options Δλ: 0.00 4.00 4.03 4.32 4.11 5.07 RooFit default 18/20
-1.91
+3.16 6
+2.00
+2.35
+3.16
+2.40 5 4 3
-2.00
-1.68
-1.16
-1.71 2 1
Nothing
Poisson
Likelihood
Bayesian central
Bayesian equal prob
Frequentist
Δλ:
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19. Discussions in other experiment:
Example  discussion in CDF:
We feel it is important to have a relatively simple rule that is readily understood by readers. A reader does not want to have to work hard simply to understand what an error bar on a plot represents. <…> Since the use of +-sqrt(n) is so widespread, the argument in favour of an alternative should be convincing in order for it to be adopted.
19/20

20. Summary
di-muon invariant mass [MeV/c2]
Candidates (50 MeV/c2)
+3.16 4
-1.91
- You now know how Roofit ‘Poisson errors’ are defined Note: choice has no impact on likelihood fits - Do you agree with RooFit default? What about empty bins then?
- Perfect topic to confuse and irritate people over coffee  do it!