Statistical methods constitute a fundamental tool for the analysis and interpretation of experimental physics data. In this course, students will learn the foundations of statistical data analysis methods and how to apply them to the analysis of experimental data. Problem sets will include computer simulations and analyzing data-sets from real experiments that require the use of programming tools to extract physics results. Topics include probability and statistics, experimental uncertainties, parameter estimation, confidence intervals, and hypothesis testing. Students will be required to complete a final project.
We believe that a solid and rigorous foundation in statistical methods is essential for the formation of an experimental physicist. Statistical methods are crucial for optimal design of physics detectors and experiments, analyses of physics data, and interpretation and communication of results in the form of new measurements, limits on searches for new phenomena, or discoveries.
The ultimate goal of this course is to provide students with a strong foundation in probability and statistics and their applications to the analysis and interpretation of physics experiments. Students will learn both theoretical concepts and computational techniques and practice thinking about data like physicists. Furthermore, this solid foundation will also prepare students for learning more advanced topics like machine learning, often required given the complexity and the scale of modern and future physics experiments. Our emphasis as instructors is to encourage a deep understanding of statistical concepts and their connections to physic research. We try to connect every statistical concept to real physics examples and computer simulations.
Basics of probability theory: Random variables. Conditional probability. Independence and correlation. Bayes’ theorem. Mean value and variance. Probability and density distribution. Multivariate probability densities and covariance.
Important probability distributions: Definition and applications of the Binomial, Poisson, Uniform, Normal, Exponential, Gamma, Beta, and Cauchy distributions. Binormal distribution and covariance ellipse. Sampling distributions: Chi-squared, Student’s t. Characteristic function. Central limit theorem.
Measurement errors: Functions of random variables. Types of measurement uncertainty: statistical and systematic errors. Error propagation. Averaging of uncorrelated and correlated measurements.
Theory of Estimators: General properties: consistency, convergence, bias, efficiency, sufficiency. Accuracy and Precision. Fisher Information. Lower bound for the variance: Cramer-Rao inequality. Maximum likelihood estimators. The least squares method. Linear regression. The least square method with errors in both variables and with non-linear functions.
Confidence intervals: Frequentist confidence intervals. Covariance ellipse. The Neyman belt construction. Upper and lower limits. Bayesian intervals. Frequentist and Bayesian interpretation of probability. Priors and Posteriors.
Hypothesis testing: General properties. Critical region and significance level. Type I, II errors. Power of a test. p-values. The Neyman-Pearson test.