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I need to study Markov Chain Monte Carlo methods, to be more specific I need to study Metropolis Hastings algorithm and all about it like convergence criteria.

Who can prescribe me a book, or a paper, or a web site, that explain this argument using simple terms, but without being trivial?

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An excelent introductory paper is Chib and Greenberg's

Unnderstanding the Metropolis-Hasting Algorithm

A masterful and concise discussion of the theory is Tierney's

Markov Chains for Exploring Posterior Distributions

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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. $\endgroup$ – EngrStudent 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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Here is a crude analogy I have used to roughly give the flavor of MHA: Next time you're at supermarket:

  1. Grab an item at random and put in your cart.

  2. Grab another item with your right hand.

  3. If item in your hand is priced less than the last item you carted, put it in your cart.

  4. Otherwise place item in your cart with probability (price of last) ÷ (price in hand) else reshelve it.

  5. Repeat steps 2 thru 4 until twenty-nine additional items are in your cart.

  6. Remove first 15 items from your cart.

  7. Checkout and wish cashier a pleasant day.

  8. Roll cart to your car.

  9. Drive home.

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