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Markov chains with general state space, geometric ergodicity.
Gibbs sampler, Metropolis-Hastings algorithm, properties and applications.
Last update: T_KPMS (19.04.2016)
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The course should give insight into the basics of Markov chains with general state space which are necessary for understanding the theoretical properties of MCMC methods. Students should become familiar with commonly used MCMC algorithms and after the course they should be able to apply those algorithms to problems in Bayesian and spatial statistics. Last update: T_KPMS (16.05.2013)
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The course is finalized by a credit from exercise class and by a final exam.
The credit from exercise class is necessary for taking part in the final exam.
Requirements for receiving the credit from exercise class:
1) active participance in the exercise class and handing in the 5 assigned homework exercises
and
2) solution of the credit homework assignment (includes theoretical analysis and practical implementation of an MCMC algorithm for a particular problem).
Attempt to receive the credit from exercise class cannot be repeated. Last update: Prokešová Michaela, RNDr., Ph.D. (05.10.2022)
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S. Brooks, A. Gelman, G. L. Jones, X. Meng (2011): Handbook of Markov Chain Monte Carlo, Chapman & Hall/CRC, Boca Raton.
D. Gamerman a H. F. Lopes (2006): Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, 2nd ed., Chapman & Hall/CRC, Boca Raton.
W. S. Kendall, F. Liang, L.-S. Wang (Eds.) (2005): Markov Chain Monte Carlo: Innovations and Applications, World Scientific, Singapore.
S. P. Meyn a R. L. Tweedie (2009): Markov Chains and Stochastic Stability, 2nd ed., Cambridge University Press, Cambridge.
C. P. Robert (2001): The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation, druhé vydání, Springer, New York. Last update: Prokešová Michaela, RNDr., Ph.D. (08.10.2015)
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Lecture+exercises. Last update: T_KPMS (16.05.2013)
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The final exam is oral. All material covered during the course may be part of the exam. Last update: Prokešová Michaela, RNDr., Ph.D. (01.08.2018)
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1. Examples of simulation methods. 2. Bayesian statistics, hierarchical models. 3. Examples of MCMC algorithms, Gibbs sampler, Metropolis-Hastings algorithm. 4. Theory of Markov chains with general state space. 5. Ergodicity of MCMC algorithms. 6. Practical aspects and estimation of limit variance. 7. Metropolis-Hastings-Green algorithm. 8. Point processes, birth-death Metropolis-Hastings algorithm. 9. Further applications. Last update: Prokešová Michaela, RNDr., Ph.D. (25.09.2020)
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Conditional probability and conditional mean value, discrete time Markov chains with discrete state-space including stationary and limit distributions and ergodicity theorem for these Markov chains. Last update: Prokešová Michaela, RNDr., Ph.D. (30.05.2018)
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