Python is one of the most versatile and widely used scripting languages. Its clean and uncluttered syntax as well as its straightforward design greatly contribute to this success and make it an ideal language for programming education. Its application ranges from web development to scientific computing. Especially in the fields of data science and artificial intelligence, it is the most common programming language supported by all major data-handling and analytical frameworks. This course provides a thorough introduction to the language and its main features, as well as insights into the rationale and application of important adjacent concepts such as environments, testing, and version control.

1. Introduction to Python
   1. Data structures
   2. Functions
   3. Flow control
   4. Input / Output
   5. Modules & packages
2. Classes and inheritance
   1. Scopes and namespaces
   2. Classes and inheritance
   3. Iterators and generators
3. Errors and exceptions
   1. Syntax errors
   2. Handling and raising exceptions
   3. User-defined exceptions
4. Important libraries
   1. Standard Python library
   2. Scientific calculations
   3. Speeding up Python
   4. Visualization
   5. Accessing databases
5. Working with Python
   1. Virtual environments
   2. Managing packages
   3. Unit and integration testing
   4. Documenting code
6. Version control
   1. Introduction to version control
   2. Version control with GIT

Modern techniques to analyze data and derive predictions for future events are deeply rooted in mathematical techniques.

The course builds a solid base to understand the concepts behind advanced algorithms used to process, analyze, and predict data and observations and enables students to follow future research, especially in the fields of data-intensive sciences.

The course reviews differentiation and integration and then discusses partial differentiation, differentiation, vector algebra and vector calculus. Matrix calculation and vector spaces are fundamental to many modern data processing algorithms and are discussed in detail. Calculations based on Tensors are introduced.

Common metrics are discussed from an informational, theoretical point of view.

1. Calculus
   1. Differentiation & Integration
   2. Partial Differentiation & Integration
   3. Vector Analysis
   4. Calculus of Variations
2. Integral Transformations
   1. Convolution
   2. Fourier Transformation
3. Vector Algebra
   1. Scalars and Vectors
   2. Addition, Subtraction of Vectors
   3. Multiplication of Vectors, Vector Product, Scalar Product
4. Vector Calculus
   1. Integration of Vectors
   2. Differentiation of Vectors
   3. Scalar and Vector Fields
   4. Vector Operators
5. Matrices and Vector Spaces
   1. Basic Matrix Algebra
   2. Determinant, Trace, Transpose, Complex, and Hermitian Conjugates
   3. Eigenvectors and Eigenvalues
   4. Diagonalization
   5. Tensors
6. Information Theory
   1. MSE
   2. Gini Index
   3. Entropy, Shannon Entropy, Kulback Leibler Distance
   4. Cross Entropy

Nearly all processes in nature and technical or scientific scenarios are not deterministic but stochastic. Therefore, these processes must be described in terms of probabilities and probability density distributions.

After defining and introducing the fundamental concepts of statistics, the course will cover important probability distributions and their prevalence in application scenarios; discuss descriptive techniques to summarize and visualize data effectively; and discuss the Bayesian approach to statistics.

Estimating parameters is a key ingredient in optimizing data models, and the course will give a thorough overview of the most important techniques.

Hypothesis testing is a crucial aspect in establishing the observation of new effects and determination of the significance of statistical effects. Special focus will be given to the correct interpretation of p-Values and the correct procedure for multiple hypothesis tests.

1. Introduction to Statistics
   1. Random Variables
   2. Kolmogorov Axioms
   3. Probability Distributions
   4. Decomposing probability distributions
   5. Expectation Values and Moments
   6. Central Limit Theorem
   7. Sufficient Statistics
   8. Problems of Dimensionality
   9. Component Analysis and Discriminants
2. Important Probability Distributions and their Applications
   1. Binomial Distribution
   2. Gauss or Normal Distribution
   3. Poisson and Gamma-Poisson Distribution
   4. Weibull Distribution
3. Bayesian Statistics
   1. Bayes’ Rule
   2. Estimating the Prior, Benford’s Law, Jeffry’s Rule
   3. Conjugate Prior
   4. Bayesian & Frequentist Approach
4. Descriptive Statistics
   1. Mean, Median, Mode, Quantiles
   2. Variance, Skewness, Kurtosis
5. Data Visualization
   1. General Principles of Dataviz/Visual Communication
   2. 1D, 2D Histograms
   3. Box Plot, Violin Plot
   4. Scatter Plot, Scatter Plot Matrix, Profile Plot
   5. Bar Chart
6. Parameter Estimation
   1. Maximum Likelihood
   2. Ordinary Least Squares
   3. Expectation Maximization (EM)
   4. Lasso and Ridge Regularization
   5. Propagation of Uncertainties
7. Hypothesis Test
   1. Error of 1st and 2nd Kind
   2. Multiple Hypothesis Tests
   3. p-Value

Statistical inference and causal analysis are crucial tools for analyzing and understanding data on a fundamental level. This course starts with an introduction to Bayesian inference and Bayesian networks which use probabilities to describe statistical problems and introduce probabilistic modelling which allows the specification of Bayesian statistical models in code.

This course introduces the concepts of causality, how causality relates to correlation between variables, and discusses the fundamental building blocks of causal analysis. The effect of interventions (i.e., when the experimenter actively changes the setup from which the data are taken) are also discussed. This course then introduces the rules of do-calculus, which allow interventions to be described formally.

Finally, the course discusses a wide range of typical fallacies which arise in the context of causal analysis.

1. Statistical Inference
   1. Bayesian inference
   2. Bayesian networks
   3. Probabilistic modelling
2. Introduction to Causality
   1. Correlation vs causation
   2. Granger causality
   3. Directed Acyclic Graphs (DAG)
   4. Elements of causal graphs: collider, chain, fork
   5. D – separation
3. Interventions
   1. Seeing vs doing
   2. Conditional independence
   3. Confounders & counterfactuals
   4. Causal inference vs randomized controlled trials
4. Do-calculus
   1. Front- & backdoor criterion
   2. Three rules of do-calculus
5. Fallacies
   1. Mediation fallacy
   2. Collider bias
   3. Simpson’s & Berkson’s Paradox
   4. Imputing missing values: causal vs data-driven view