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I have fairly good practical experience with Metropolis-Hastings and Gibbs sampling, but I want to get a better mathematical understanding of these algorithms. What are some good textbooks or articles that prove the correctness of these samplers (more algorithms would also be great)?

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For a handbook and an extensive coverage, the following one is very moderately priced.

Brooks, et al. (ed.), Handbook of Markov Chain Monte Carlo, Chapman & Hall/CRC, 2011.

Robert and Casella (2010) have a good deal of theory.

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I'm not sure whether this is exactly what you're after, but a couple of articles I've found useful on theoretical properties of various Metropolis-Hastings algorithms are:

The book by Robert & Casella (mentioned above) is a very good and thorough resource, but you may also find these two of use:

These both also have information on Gibbs sampling. I suppose you may also find some other information on Markov Chains useful. I generally use:

But there is also some good information in most MCMC books on this.

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  • $\begingroup$ +1 Welcome to our site, Sam! Thank you for offering such an informative reply. $\endgroup$ – whuber Sep 3 '12 at 14:31

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