Textbook deriving Metropolis-Hastings and Gibbs Sampling

Question

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)?

Answer 1

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.

Answer 2

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:

• Optimal scaling for various Metropolis-Hastings algorithms - Roberts & Rosenthal, 2001.

  (This summarises some earlier results for the Ransom walk Metropolis and the Metropolis-adjusted Langevin algorithm.)

• The Random Walk Metropolis: linking theory and practice through a case study - Sherlock, Fearnhead & Roberts, 2009

  (This may be a good bridge between theoretical properties and practical use, as suggested by the title.)

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

• Markov Chain Monte Carlo in Practice - Gilks, Richardson & Spiegelhalter (1995)

• Markov Chain Monte Carlo: Stochastic simulation for Bayesian inference - Gamerman (2006)

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

• Markov Chains - Norris, (1998).

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