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The first lecture of a statistical data analysis course by glen cowan from the university of london. The lecture covers the outline of the topics to be discussed in the course, including probability, bayes' theorem, random variables, probability densities, expectation values, error propagation, and the monte carlo method. The document also provides references to related statistics books and papers.
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London Postgraduate Lectures on Particle Physics; University of London MSci course PH Glen Cowan Physics Department Royal Holloway, University of London [email protected] www.pp.rhul.ac.uk/~cowan Course web page: www.pp.rhul.ac.uk/~cowan/stat_course.html
1 Probability, Bayes’ theorem 2 Random variables and probability densities 3 Expectation values, error propagation 4 Catalogue of pdfs 5 The Monte Carlo method 6 Statistical tests: general concepts 7 Test statistics, multivariate methods 8 Goodness-of-fit tests 9 Parameter estimation, maximum likelihood 10 More maximum likelihood 11 Method of least squares 12 Interval estimation, setting limits 13 Nuisance parameters, systematic uncertainties 14 Examples of Bayesian approach
Observe events of a certain type Measure characteristics of each event (particle momenta, number of muons, energy of jets,...) Theories (e.g. SM) predict distributions of these properties
F
Z
s , m H
Some tasks of data analysis: Estimate (measure) the parameters; Quantify the uncertainty of the parameter estimates; Test the extent to which the predictions of a theory are in agreement with the data.
In particle physics there are various elements of uncertainty: theory is not deterministic quantum mechanics random measurement errors present even without quantum effects things we could know in principle but don’t e.g. from limitations of cost, time, ... We can quantify the uncertainty using PROBABILITY
Also define conditional probability of A given B (with P(B) ≠ 0 ): E.g. rolling dice: Subsets A, B independent if: If A, B independent, N.B. do not confuse with disjoint subsets, i.e.,
I. Relative frequency A , B , ... are outcomes of a repeatable experiment cf. quantum mechanics, particle scattering, radioactive decay... II. Subjective probability A , B , ... are hypotheses (statements that are true or false)
Consider a subset B of the sample space S , B ∩ A i
i
divided into disjoint subsets A i such that i
i
→ law of total probability Bayes’ theorem becomes
Suppose the probability (for anyone) to have AIDS is: ← prior probabilities, i.e., before any test carried out Consider an AIDS test: result is + or ← probabilities to (in)correctly identify an infected person ← probabilities to (in)correctly identify an uninfected person Suppose your result is +. How worried should you be?
In frequentist statistics, probabilities are associated only with the data, i.e., outcomes of repeatable observations (shorthand: ). Probability = limiting frequency Probabilities such as P (Higgs boson exists),
s
etc. are either 0 or 1, but we don’t know which. The tools of frequentist statistics tell us what to expect, under the assumption of certain probabilities, about hypothetical repeated observations. The preferred theories (models, hypotheses, ...) are those for which our observations would be considered ‘usual’.
In Bayesian statistics, use subjective probability for hypotheses: posterior probability, i.e., after seeing the data prior probability, i.e., before seeing the data probability of the data assuming hypothesis H (the likelihood) normalization involves sum over all possible hypotheses Bayes’ theorem has an “if-then” character: If your prior
should change in the light of the data. No general prescription for priors (subjective!)
Probability to have outcome less than or equal to x is cumulative distribution function Alternatively define pdf with
pdf = histogram with infinite data sample, zero bin width, normalized to unit area.
Sometimes we want only pdf of some (or one) of the components: → marginal pdf x 1 , x 2 independent if i
Marginal pdf ~ projection of joint pdf onto individual axes.
