What are some good references for the Metropolis-Hastings algorithm? 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?
 A: For a book that is not "heavy on the math", I'd recommend: 


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*Doing Bayesian Data Analysis: A Tutorial with R and BUGS by John K. Kruschke.


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.
A: 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.
A: 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
A: 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. 
