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I need to study the Metropolis-Hastings algorithm and its properties, like convergence criteria. What is a good book, paper, or website that explains it using simple terms, but without being trivial?

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An excellent introductory paper is

Chib, Siddhartha, and Edward Greenberg. “Understanding the Metropolis-Hastings Algorithm.” The American Statistician, vol. 49, no. 4, 1995, pp. 327–335.

Free download

A masterful and concise discussion of the theory is

Tierney, Luke. “Markov Chains for Exploring Posterior Distributions.” The Annals of Statistics, vol. 22, no. 4, 1994, pp. 1701–1728.

Free download

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  • $\begingroup$ Thanks a lot. My principal target is to learn about convergence criteria, but i know only the base of Metropolis Hastings, thus all it's useful. $\endgroup$
    – Neptune
    May 31 '14 at 22:35
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    $\begingroup$ Start studying convergence with Tierney. An exhaustive treatment is found in Meyn and Tweedie probability.ca/MT $\endgroup$
    – Zen
    May 31 '14 at 23:18
  • $\begingroup$ And what about simulated annealing with Metropolis Hastings? I have read this but what about the integration with Metropolis Hastings? $\endgroup$
    – Neptune
    May 31 '14 at 23:26
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    $\begingroup$ Robert and Casella's book discusses simulated annealing. amazon.com/Monte-Statistical-Methods-Springer-Statistics/dp/… $\endgroup$
    – Zen
    Jun 1 '14 at 2:39
  • $\begingroup$ "Understanding..." link is broken. $\endgroup$ Feb 15 '16 at 18:53
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For a book that is not "heavy on the math", I'd recommend:

Go to Chapter 7.

R code is provided in the book, so you'll be able to play around with the examples and see, hands-on, the effects of changing the number of burn-ins and so on.

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There is a very good paper by Christian Robert describing M-H algorithm in detail

Robert, C. P. (2015). The Metropolis-Hastings algorithm. arXiv preprint arXiv:1504.01896.

and great book about Monte Carlo methods in general from the same author

Robert, C., & Casella, G. (2013). Monte Carlo statistical methods. Springer Science & Business Media.

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With regards to convergence criteria most work is on convergence is the Total Variation (TV) distance sense. Mostly because there is a lot of probability theory worked out for TV distance. There is a nice survey paper and also on the theoretical side there is the paper by Roberts and Rosenthal that gives several theorems on convergence criteria. On the more practical side there are several papers written by Jim Hobert that provide examples of applying one of the theorems in Roberts and Rosenthal to MCMC. In general the tricky part of applying that theorem seems to be coming up with a good Lyapunov drift function.

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