Dates

General information

Description

This course serves as an introduction to foundational aspects of modern statistical data analysis. Frequentist and Bayesian inference are presented from the perspective of probabilistic modeling. The course will consist of three main parts:

1. Probability foundations: probability spaces, random variables, distribution of a random variable, expectation and covariance, important limit theorems and inequalities.
2. Frequentist inference: point estimators, confidence intervals, hypothesis testing.
3. Bayesian inference: conjugate inference, numerical models, data assimilation.

Prerequisites

Basic set theory (inclusion, union, intersection, difference of sets), basic analysis (infinite series, calculus), matrix algebra, some knowledge of probabilistic foundations (discrete probability, Gaussian distributions) is helpful.

Completing the course

The conditions for completing this course are (1) successfully completing at least 60% of the course's exercises, (2) presenting at least 2 solutions to exercise problems in the exercise sessions, and (3) successfully passing the course exam.

Registration

Please register to the course via Campus Management (CM), then you will be automatically registered in MyCampus/Whiteboard as well. Please note the deadlines indicated there. For further information and in case of any problems, please consult the Campus Management's Help for Students.

Lecture notes

Lecture notes will be published here after each week's lecture.

• Week 1: Introduction, probability space, conditional probability, independence of events
• Week 2: Random variables
• Week 3: Joint distributions, independent random variables, conditional distribution, transformations of random variables
• Week 4: Expected value and covariance
• Week 5: Inequalities and limits
• Week 6: Introduction to statistical inference, descriptive statistics, confidence interval
• Week 7: Hypothesis testing: t-tests, variance tests, nonparametric tests
• Week 8: Proportion tests, normality tests, chi-squared tests
• Week 9: Correlation and dependence in statistics, linear regression
• Week 10: Tests and confidence intervals for linear regression, multivariate linear regression, maximum likelihood estimator
• Week 11: Brief overview of Bayesian inference (files: source.py, priormodeling.py, cauchy.py)
• Week 12: Linear Gaussian setting and the Kalman filter (files: deconv.py)
• Week 13: Markov Chain Monte Carlo (files: mh.py, gibbs.py)

Brief summary of probability theory (weeks 1–3)

Exercise sheets

Weekly exercises will be published here after each lecture.

• Exercise 1
• Exercise 2
• Exercise 3
• Exercise 4
• Exercise 5
• Exercise 6 (files: mtcars.txt, mtcars.xlsx, HW.txt, FT.txt, BP2.txt)
• Exercise 7 (files: height.txt, FT2.txt)
• Exercise 8 (files: iris.txt, mtcars.txt, patients.txt)
• Exercise 9
• Exercise 10 (files: shoeheight.txt)
• Exercise 11
• Exercise 12
• Exercise 13
• Exercise 14 (files: signal.mat)

Contact

Literature

Lecture notes of the WS22 course written by Henri Elad Altman

• Larry Wasserman. All of Statistics: A Concise Course in Statistical Inference. Springer Science & Business Media, 2004.
• Morris H. DeGroot and Mark J. Schervish. Probability and Statistics. 4th edition, Pearson Education, 2013.
• José M. Bernardo and Adrian F. M. Smith. Bayesian Theory. 2nd edition, Wiley, 2007.
• Leonhard Held and Daniel Sabanés Bové. Applied Statistical Inference: Likelihood and Bayes. Springer Science & Business Media, 2013.
• Sebastian Reich and Colin Cotter. Probabilistic Forecasting and Bayesian Data Assimilation. Cambridge University Press, 2015.