Bayesian principles in statistics

Garant předmětu:

Václav Kůs

Lecturer:

Václav Kůs

Tutor:

Supervisor:

Department of Mathematics

Synopsis:

The main goal of the subject is to provide decision making mathematical principles with random effects, optimal and robust strategies and their mutual links together with computational aspects for the real applications. The techniques are illustrated within practical examples originating from point and interval estimation and statistical hypothesis testing.

Requirements:

Basic course of Calculus and Probability - in the extent of the courses 01MAA3-4 or 01MAB3-4, 01MIP or 01PRST.

Syllabus of lectures:

1. Sufficient statistics, general principles of classical statistics, conditionality, likelihood, sequential principles and their relations, Bayesian principle, Bayesian complete model and its advantages.

2. Loss and risk functions, utility function and its existence, general decision functions. Optimal decision and complete classes of optimal strategies.

3. Convex loss functions, Rao-Blackwell theorem, uniformly best strategy, unbiasness, UMVUE construction, examples.

4. Bayes optimal decision strategy, prior and posterior Bayesian risk. Families of aprior informations, uncertainty principle.

5. Jeffreys densities, conjugated systems, limit aposteriory densities, examples for standard families.

6. Minimax strategies, admissibility principle and its consequences within classical and Bayesian statistics, Stein effect.

7. Score functions and their robust properties, Shannon entropy, f-divergences, maximum entropy principle, new extended families of divergences and its metric and robust properties.

8. Minimum distance point estimators, minimum Kolmogorov, Lévy and discrepancy decision functions and its L1-consistency and qualitative robustness, Kolmogorov entropy, Vapnik-Chervonenkis dimension.

9. Numerical procedures, approximative calculations in higher dimensions, Monte-Carlo approaches, importance sampling, convergence, Metropolis algorithm.

10. Second order Laplace asymptotic expansion, fully exponential forms, regularity assumptions for stochastic expansion/approximation, the results of Kass-Tierney-Kadaneho.

11. Hierarchic Bayes, Empirical Bayes, Variational Bayes - principles and examples.

12. Bayesian hypothesis testing for various loss functions, properties.

Syllabus of tutorials:

Study Objective:

Knowledge:

Extension of the decision makinng principles with random effects and their application in stochastic optimization tasks, mainly in Bayesian methods.

Skills:

Orientation in various stochastical approaches and their properties. Computational aspects.

Study materials:

Key references:

[1] Berger J.O., Statistical Decision Theory and Bayesian Analysis, Springer, N.Y., 1985.

[2] Maitra A.P., Sudderth W.D., Discrete Gambling and Sochastic Games, Springer, 1996.

Recommended references:

[3] Fishman G.S., Monte Carlo, Springer, 1996.

[4] Bernardo J.M., Smith A.F.M., Bayesian Theory, Wiley, 1994.

Note:

Time-table for winter semester 2023/2024:

Time-table is not available yet

Time-table for summer semester 2023/2024:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikované matematicko-stochastické metody (compulsory course in the program)