Stochastic Games and Bayesian Decisions Making

Garant předmětu:

Lecturer:

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, 01PRA1 nebo 01PRST held at the FNSPE CTU in Prague).

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, decision functions, . convex loss functions, Rao-Blackwell theorem, complete classes of optimal strategies, conditions. Bayes optimal decision strategy, prior and posterior Bayesian risk. Families of aprior informations, uncertainty principle, Jeffreys densities, conjugated systems, examples. 3. Minimax strategies, uniformly best strategy, minimum distance strategy, admissibility principle and its consequences within classical and Bayesian statistics.Stein effect. 4. Score functions and their robust properties, Shannon entropy, f-divergences, maximum entropy principle, new extended families of divergences and its metric and robust properties, minimum distance point estimators, min.Kolmogorov, Lévy and discrepancy decision functions and its L1 consistency and qualitative robustness, Kolmogorov entropy, Vapnik-Chervonenkis dimension. 5. Numerical procedures, approximative calculations in higher dimensions, Monte-Carlo approaches, importance sampling, convergence, Metropolis algorithm. Second order Laplace asymptotic expansion, fully exponential forms, regularity assumptions for stochastic expansion/approximation, the results of Kass-Tierney-Kadaneho.

Syllabus of tutorials:

Principle calculations for the specific optimal strategies and their properties: 1. Sufficient statistics, likelihood methods, applications of the Rao-Blackwell theorem, calculations of the best unbiased decision functions for Poisson distribution. 2. Bayesian principle within various loss functions, prior distribution selection, calculations for the uncertainty principle, Jeffreys densities and conjugated families of prior densities for known distributions. 3. Minimax strategy, James-Steinovy shrinkage estimates. 4. Minimum distance decisions, f-divergence calculations for particular distributions, Vapnik-Chervonenkis dimension illustrations. 5. Monte Carlo optimal decision in case of normal truncated model.

Study Objective:

Knowledge: Extension of the decision makinng and game theory principles with random effects and their application in stochastic optimization tasks.

Abilities: Orientation in various stochastical approaches and their properties.

Study materials:

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

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

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

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

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: