Bayesian statistics

elective courses

Systematic introduction to Bayesian statistics. The subject of this course is now becoming more popular, has many important

applications, but is treated marginally or entirely omitted in standard courses of statistics. The course is dedicated to students of mathematics and also students of informatics who are interested in statistics.

1. Classical and Bayesian viewpoint in statistics. Probabilistic foundations: conditional probability distributions, conditional expectations, Bayes formula.

2. Constructing Bayesian models. Prior and posterior distributions. Predicive distributions. Conditional independence and sufficiency. Conjudate families of distributions. Standard examples of Bayesian models.

3. Loss functions, Bayesian estimation and prediction. Basics of statistical decision theory. Applications: classification, pattern recognition, mixed linear models in credibility theory and in small area estimation. Empirical Bayesian approach and hierarchical models.

4. Computational methods in Bayesian statystics. MCMC (Markov chain Monte Carlo) and SMC (sequential Monte Carlo).

Applications to hierarchical models and (hidden Markov models).

5. Hypotheses testing in Bayesian world. Bayes factors, model choice.

6. Elements of Bayesian asymptotic theory. Consistency and asymptotic normality of posterior distributions. Exchangeability and de Finetti theorem.

1. M.H. DeGooot, Optimal statstical decisions. Wiley 2004.

2. S.D. Silvey, Statistical inference. Chapman and Hall 1970.

3. C.P. Robert, The Bayesian choice: a decision-theoretic motivation. Springer 1994.

4. J.H. Albert, Bayesian computation with R. Springer 2008.

Completion of exercises on the basis of homework solutions presented during the exercises.

Final grade on the basis of a written exam, consisting of about 6 tasks. In exceptional cases, the possibility of taking an additional oral examination.