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Statistics in Particle Physics. 1. 20-29 November 2006 Tatsuo Kawamoto ICEPP, University of Tokyo. Outline. Introduction Probability Distributions Fitting and extracting parameters Combination of measurements Errors, limits and confidence intervals Likelihood, ANN, and sort of things.

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slide1

Statistics in Particle Physics

1

20-29 November 2006

Tatsuo Kawamoto

ICEPP, University of Tokyo

slide2

Outline

Introduction

Probability

Distributions

Fitting and extracting parameters

Combination of measurements

Errors, limits and confidence intervals

Likelihood, ANN, and sort of things

slide3

References

  • Textbooks of statistics in HEP
  • PDG review (Probability, Statistics)
  • Relevant scientific papers
slide5

Why bother statistics ?

  • It’s not fundamental.
  • As soon as we come to the point to present results of
  • an experiment, we face to a few questions like:
  • What is the size of uncertainty?
  • How to combine results from different runs?
  • Discovered something new?
  • If not discovery, what we can say from the experiment?
  • Prescriptions to these problems often involve considerations
  • based on statistics.
slide6

Particle Physics

Study of elementary particles that have been discovered

- Quarks

- leptons

- Gauge bosons

- Hadrons

And anything that has not been discovered

- Higgs

- Supersymmetry

- Extradimensions

slide7

Goals of experiments

For each particle we want to know, eg.

  • What are its properties ?

- mass, lifetime, spin, ….

  • What are its decay modes ?
  • How it interacts with other particles ?
  • Does it exist at all ?

Observation is a result of fundamental rules of the nature

these are random, quantum mechanical, processes

slide8

Also, the detector effects (resolution, efficiency, …) are

often of random nature

Systematic uncertainty is a subtle subject, but we have to do

our best to say something about it, and treat it reasonably.

slide9

Template for an experiment

  • To study X
  • Arrange for X to occur
  • e.g colliding beams
  • Record events that might be X
  • trigger, data acquisition,
  • Reconstruct momentum, energy, … of visible particles
  • Select events that could be X by applying CUTS

Efficiency < 100%, Background > 0

  • Study distributions of interesting variables
  • Compare with/ fit to Theoretical distributions
  • Infer the value of parameter q and its uncertainty
slide10

Implications

  • Essentially counting numbers
  • Uncertainties of measurements are understood
  • Distributions are reproduced to reasonable accuracy
slide11

We don’t use:

  • Student’s t
  • F test
  • Markov chains
slide12

Tools

  • Monte Carlo simulation
  • Know in principle → Know in practice
  • Simple beautiful underlying physics
  • Unbeautiful effects (higher order, fragmentation,..)
  • Ugly detector imperfections (resolution, efficiency)
  • Likelihood
  • Fundamental tool to handle probability
  • Fitting
  • c2, Likelihood, Goodness of fit
  • Toy Monte Carlo
  • Handle complicated likelihood
slide13

Extracting parameters

Example:

mZ = 91.1853±0.0029 GeV

GZ = 2.4947 ±0.0041 GeV

shad= 41.82 ±0.044 nb

slide17

Likelihood, Artificial Neural Net

Use as much

Information as

possible

Example:

W+W- → qqqq

slide18

There are other important things

which we don’t cover

  • Blind analysis
  • Unfolding
  • ….
slide19

2. Probability

What is it?

slide20

Mathematical

P(A) is a number obeying the rules:

Kolmogorov axioms

Ai are disjoint events

slide21

Mathematical

Lemma

And, that’s almost it.

slide22

Classical

Laplace, …

From considerations of games of chances

Given by symmetry for equally-likely outcomes, for which

we are equally undecided.

Classify things into certain number of equally-likely cases,

And count the number of such favorable cases.

P(A) = number of equally-likely favorable cases / total number

Tossing a coin P(H)=1/2, Throwing a dice P(1)=1/6

How to handle continuous variables ?

slide23

Frequentist

Probability is the limit of frequency (taken over some ensemble)

The event A either occur or not. Relative frequency of occurence

Law of large numbers

slide25

Frequency definition is associated to some ensemble of ‘events’

Can’t say things like:

  • It will probably rain tomorrow
  • Probability of LHC collision in November 2007
  • Probability of existence of SUSY

But one can say:

  • The statement ‘It will rain tomorrow’ is probably true

Comeback later in the discussion of confidence level

slide26

Bayesian or Subjective probability

P(A) is the degree of belief in A

A can be anything:

Rain, LHC completion, SUSY, ….

You bet depending on odds P vs 1-P

slide27

Bayes theorem

Often used in subjective probability discussions

Conditional probability P(A|B)

Thomas Bayes 1702-1761

slide28

Bayes theorem

How it works?

Initial belief P(Theory) is modified by experimental results

If Result is negative, P(Result|Theory)=0, the Theory is killed

P(Theory|Result)=0

It’s an extreme case. Will comeback later in the discussion of

confidence level

slide29

Fun with Bayes theorem - 1

Monty Hall problem

  • There are 3 doors
  • Behind one of these, there is a prize (a car, etc)
  • Behind each of the other two, there is a goat (you lost)
  • you choose 1 door whatever you like (you bet), say, Nr 1.
  • A door will be opened to reveal a goat, either of Nr 2 or Nr 3,
  • chosen randomly if goat is behind the both.
  • Then you are asked if you stay Nr 1, or, switch to Nr 2.

You should stay or switch?

slide30

One would say:

you don’t know anyway if there is the prize

behind Nr 1 or Nr 2. They are equally probable.

To stay or to switch give equal chance.

slide31

But the correct strategy is to switch

A ‘classical’ reasoning (count the number of cases)

Before the door is opened

After the door is opened

Odds to win : stay 1/3

switch 2/3

slide32

Using Bayes theorem

P(Ci) : Prize is behind door i = 1/3

P(Ok) : Door k is opened

We want to know P(C1| O3) vs P(C2| O3)

slide33

Exercise

A disease X (maybe AIDS, SARS, ….)

P(X) = 0.001 Prior probability

P(no X) = 0.999

Consider a test of X

P(+ | X) = 0.998

P(+ | no X) = 0.03

If the test result were +, how worried you should be ?

ie. What is P(X | +) ?