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Markov Chain Monte Carlo

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Numerical Analysis for Statisticians

Part of the book series: Statistics and Computing ((SCO))

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Abstract

The Markov chain Monte Carlo (MCMC) revolution sweeping statistics is drastically changing how statisticians perform integration and summation. In particular, the Metropolis algorithm and Gibbs sampling make it straightforward to construct a Markov chain that samples from a complicated conditional distribution. Once a sample is available, then any conditional expectation can be approximated by forming its corresponding sample average. The implications of this insight are profound for both classical and Bayesian statistics. As a bonus, trivial changes to the Metropolis algorithm yield simulated annealing, a general-purpose algorithm for solving difficult combinatorial optimization problems.

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References

  1. Besag J, Green PJ (1993) Spatial statistics and Bayesian computation. J Roy Stat Soc B 55:25-37

    MATH  MathSciNet  Google Scholar 

  2. Brualdi RA (1977) Introductory Combinatorics. North-Holland, New York

    MATH  Google Scholar 

  3. Casella G, George EI (1992) Explaining the Gibbs sampler. Amer Statistician 46:167-174

    Article  MathSciNet  Google Scholar 

  4. Castledine B (1981) A Bayesian analysis of multiple-recapture sam- pling for a closed population. Biometrika 67:197-210

    Article  MathSciNet  Google Scholar 

  5. Calvetti D, Somersalo E (2007) Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing. Springer, New York

    MATH  Google Scholar 

  6. Chib S, Greenberg E (1995) Understanding the Metropolis-Hastings algorithm. Amer Statistician 49:327-335

    Article  Google Scholar 

  7. Duane S, Kennedy AD, Pendleton BJ, Roweth D (1987) Hybrid Monte Carlo. Physics Letters B 195:216-222

    Article  Google Scholar 

  8. Fishman GS (2006) A First Course in Monte Carlo. Duxbury Press, Belmont, CA

    Google Scholar 

  9. Gelfand AE, Smith AFM (1990) Sampling-based approaches to calculating marginal densities. J Amer Stat Assoc 85:398-409

    Article  MATH  MathSciNet  Google Scholar 

  10. Gelman A, Carlin JB, Stern HS, Rubin DB (2003) Bayesian Data Analysis, 2nd ed. Chapman & Hall, London

    Google Scholar 

  11. Gelman A, Rubin DB (1992) Inference from iterative simulation using multiple sequences (with discussion). Stat Sci 7:457-511

    Article  Google Scholar 

  12. Geman S, Geman D (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Trans Pattern Anal Machine Intell 6:721-741

    Article  MATH  Google Scholar 

  13. George EI, Robert CP (1992) Capture-recapture estimation via Gibbs sampling. Biometrika 79:677-683

    MATH  MathSciNet  Google Scholar 

  14. Geyer CJ (1991) Markov chain Monte Carlo maximum likelihood. Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, Keramidas EM, editor, Interface Foundation, Fairfax, VA pp 156-163

    Google Scholar 

  15. Gilks WR, Richardson S, Spiegelhalter DJ (1996) Markov Chain Monte Carlo in Practice. Chapman & Hall, London

    MATH  Google Scholar 

  16. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57:97-109

    Article  MATH  Google Scholar 

  17. Higdon DM (1998) Auxiliary variable methods for Markov chain Monte Carlo with applications. JASA 93:585-595

    MATH  Google Scholar 

  18. Jennison C (1993) Discussion on the meeting of the Gibbs sampler and other Markov chain Monte Carlo methods. J Roy Stat Soc B 55:54-56

    Google Scholar 

  19. Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671-680

    Article  MathSciNet  Google Scholar 

  20. Lange K, Sobel E (1991) A random walk method for computing genetic location scores. Amer J Hum Genet 49:1320-1334

    Google Scholar 

  21. Liu JS (1996) Metropolized independent sampling with comparisons to rejection sampling and importance sampling. Stat and Computing 6:113-119

    Article  Google Scholar 

  22. Liu JS (2001) Monte Carlo Strategies in Scientific Computing. Springer, New York

    MATH  Google Scholar 

  23. Mallows CL (1973) Some comments on Cp. Technometrics 15:661-675

    Article  MATH  Google Scholar 

  24. Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equations of state calculations by fast computing machines. J Chem Physics 21:1087-1092

    Article  Google Scholar 

  25. Neal RM (1996) Bayesian Learning for Neural Networks. Springer, New York

    MATH  Google Scholar 

  26. Neal RM (2003) Slice sampling. Ann Stat 31:705-767

    Article  MATH  MathSciNet  Google Scholar 

  27. Press WH, Teukolsky SA, Vetterling WT, Flannery BP (1992) Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. Cambridge University Press, Cambridge

    Google Scholar 

  28. Robert CP, Casella G (2004) Monte Carlo Statistical Methods, 2nd ed. Springer, New York

    MATH  Google Scholar 

  29. Spiegelhalter D, Thomas D, Best N, Lunn D (2006) Winbugs Users Manual 1.4.1, Bayesian Inference Using Gibbs Sampling. MRC Biostatistics Unit, Institute of Public Health, Cambridge, Available: http://www.mrc-bsu.cam.ac.uk/bugs

  30. Swendsen RH, Wang JS (1987) Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters 58:86-88

    Article  Google Scholar 

  31. Suchard MA, Bailey JN, Elashoff DE, Sinsheimer JS (2001) SNPing away at candidate genes. Genetic Epidemiology 21:S643-S648

    Google Scholar 

  32. Tanner MA (1993) Tools for Statistical Inference: Methods for Exploration of Posterior Distributions and Likelihood Functions, 2nd ed. Springer, New York

    MATH  Google Scholar 

  33. Tanner M, Wong W (1987) The calculation of posterior distributions (with discussion). J Amer Stat Assoc 82:528-550

    Article  MATH  MathSciNet  Google Scholar 

  34. Tierney L (1994) Markov chains for exploring posterior distributions (with discussion). Ann Stat 22:1701-1762

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departments of Biomathematics, Human Genetics, and Statistics David Geffen School of Medicine, University of California, Los Angeles, Le Conte Ave. 10833, Los Angeles, CA, 90095-1766, USA

    Kenneth Lange

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  1. Kenneth Lange
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Correspondence to Kenneth Lange .

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Lange, K. (2010). Markov Chain Monte Carlo. In: Numerical Analysis for Statisticians. Statistics and Computing. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5945-4_26

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