Statistics and Data Analysis (HEP at LHC)

 Computing statistical results

 Estimating the value of a parameter

 Testing hypotheses

 Discovery

 Limits

 Confidence intervals

Statistics and Data Analysis (HEP at LHC)

Prof. dr hab. Elżbieta Richter-Wąs

Slides extracted from N. Berger lectures at CERN Summer School 2019

How to represent the data

2

Model parameters

3

Statistical computations

4

Using the PDF

5

Likelihood

6

Poisson example

7

Poisson example

8

Poisson example

9

Maximum Likelihood Estimation

10

MLEs in shape analyses

11

MLEs in shape analyses

12

MLE Properties

13

Hypothesis Testing

14

Hypothesis Testing

15

Hypothesis Testing

16

ROC Curves

17

ROC Curves

18

ROC Curves

19

Hypothesis testing with Likelihoods

20

Discovery: Test Statistic

21

Discovery: p-value

22

Asymptotic distribution of q₀

23

Some examples

24

Takeaways

25

Hypothesis tests for Limits

26

Hypothesis tests for Limits

27

Hypothesis tests for Limits

28

Hypothesis tests for Limits

29

Test Statistic for Limit-Setting

30

Inversion: Getting the limit for a given CL

31

Inversion: Getting the limit for a given CL

32

Inversion: Getting the limit for a given CL

33

Upper Limit Pathologies

34

CL_s

35

Upper Limit Examples

36

Gaussian Intervals

37

Likelihood Intervals

38

2D examples

39

Takeaways

40

Extra slides

41

Discovery significance

42

One-sided vs. Two-sided

43

One-sided Asymptotics

44

One-sided Test Statistic

45

CL_S: Gaussian Bands

46

Comparison with LEP/Tevatron definitions

47

Spin/Parity measurements

48