Bayesian Methods

Part 1. Introduction to the Bayesian theory

Principles, uncertainty, prior/posterior knowledge, hyperparameters, likelihood, Bayes theorem, difference to frequentist statistics, sequential (online) modeling. Types of estimates.
Prior and posterior distribution, the existence of analytical solutions, conjugacy, exponential family, real-world examples
Linear regression and autoregression, types of reality-reflecting models (black box, grey box), construction of linear models based on physical principles
Basics of Bayesian estimation with conjugate prior distributions (Practical classes)
Sequential (online) linear regression and autoregression with real data. Exponential forgetting in the estimation of models with time-varying parameters (Practical classes)

Part 2. Generalized linear models

Logistic regression and online logistic regression, MAP estimation. Sketch of other GLMs. Practical examples
State-space models I
Evolution of model parameters (i.e., states). Linear state-space models, Kalman filtering. Real-world examples (Apollo program, free fall equation, target tracking)
Sequential logistic regression (Practical classes)
Kalman filtering (Practical classes)

Part 3. State-space models II/Advanced methods

Nonlinear models and introduction to particle filtering. Principles of particle filters, filter degeneracy, resampling
Discussion of principles of selected advanced methods like Dynamic Model Averaging, Approximate Bayesian Computation and/or others based on students’ interests
Particle filtering (Practical classes)
Concluding discussion