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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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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
References • Textbooks of statistics in HEP • PDG review (Probability, Statistics) • Relevant scientific papers
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.
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
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
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.
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
Implications • Essentially counting numbers • Uncertainties of measurements are understood • Distributions are reproduced to reasonable accuracy
We don’t use: • Student’s t • F test • Markov chains • …
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
Extracting parameters Example: mZ = 91.1853±0.0029 GeV GZ = 2.4947 ±0.0041 GeV shad= 41.82 ±0.044 nb
Likelihood, Artificial Neural Net Use as much Information as possible Example: W+W- → qqqq
There are other important things which we don’t cover • Blind analysis • Unfolding • ….
2. Probability What is it?
Mathematical P(A) is a number obeying the rules: Kolmogorov axioms Ai are disjoint events
Mathematical Lemma And, that’s almost it.
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 ?
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
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
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
Bayes theorem Often used in subjective probability discussions Conditional probability P(A|B) Thomas Bayes 1702-1761
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
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?
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.
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
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)
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 | +) ?
