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Markov Chain Monte Carlo on finite state spaces

Published online by Cambridge University Press:  18 June 2020

Tobias Siems
Tobias Siems*
Affiliation:
Department of Mathematics and Computer Science, University of Greifswald, Germany
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Extract

We elaborate the idea behind Markov chain Monte Carlo (MCMC) methods in a mathematically coherent, yet simple and understandable way. To this end, we prove a pivotal convergence theorem for finite Markov chains and a minimal version of the Perron-Frobenius theorem. Subsequently, we briefly discuss two fundamental MCMC methods, the Gibbs and Metropolis-Hastings sampler. Only very basic knowledge about matrices, convergence of real sequences and probability theory is required.

Type
Articles
Information
The Mathematical Gazette , Volume 104 , Issue 560 , July 2020 , pp. 281 - 287
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Copyright
© Mathematical Association 2020

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References

Bishop, C. M. and Mitchell, T. M., Pattern recognition and machine learning, Springer (2014).Google Scholar
Geman, S. and Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Transactions on pattern analysis and machine intelligence (1984).CrossRefGoogle Scholar
Hastings, W. K., Monte Carlo sampling methods using Markov chains and their applications, Biometrika (1970).CrossRefGoogle Scholar
Koenig, W., Stochastische Prozesse I: Markovketten in diskreter und stetiger Zeit University of Leipzig (2005).Google Scholar
Frobenius, G.Über Matrizen aus nicht negativen Elementen, Reimer (1912).Google Scholar
Ising, E., Contribution to the theory of ferromagnetism, Journal of Physics (1925).Google Scholar
Higdon, D. M., Auxiliary variable methods for Markov chain Monte Carlo with applications, Journal of the American Statistical Association (1998).CrossRefGoogle Scholar
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E., Equations of state calculations by fast computing machines, The Journal of Chemical Physics 21 (6) (1953) pp. 1087–1092.CrossRefGoogle Scholar