Statistical data analysis:
course outline
Statistical Data Analysis (main lectures 3:00
to 5:00):
- Probability.
Definition and interpretation, Bayes' theorem,
random variables, probability density functions,
expectation values, transformation of variables, error propagation.
- Examples of probability functions.
Binomial, multinomial,
Poisson, uniform, exponential, Gaussian, chi-square, Cauchy distributions.
- The Monte Carlo method.
Random number generators, the
transformation method, the acceptance-rejection method.
- Statistical tests.
Significance and power of a test,
choice of the critical region. Constructing test statistics: the Fisher
discriminant, neural networks, etc. Testing goodness-of-fit, chi^2-test,
P-values.
- Parameter estimation: general concepts.
Samples, estimators, bias. Estimators for mean, variance, covariance.
- The method of maximum likelihood.
The likelihood function,
ML estimators for parameters of Gaussian and exponential distributions.
Variance of ML estimators, the information inequality, extended ML,
ML with binned data.
- The method of least squares.
Relation to maximum likelihood,
linear least squares fit, LS with binned data, testing goodness-of-fit,
combining measurements with least squares.
- Interval estimation.
Classical confidence intervals: with
Gaussian distributed estimator, for mean of Poisson variable. Setting
limits, limits near a physical boundary.
- Nuisance parameters, systematic uncertainties.
Connection between systematic uncertainty and nuisance
parameters. Profile likelihood, Bayesian treatment, marginalization
with MCMC.
- Examples of the Bayesian approach.
Bayesian treatment of non-Gaussian systematic errors. Model
selection using Bayes factors.