Introductory refs for MCMC

Bayesian/Markov chain Monte Carlo (MCMC) approaches are powerful, but potentially daunting, ways to run GLMMs. They side-step a number of the technical problems with frequentist approaches.

My (BMB) suggestions for learning how to do this yourself (for now, just references):

• my book (Ecological Models and Data in R, forthcoming from Princeton University Press) puts these methods in perspective, but just scratches the surface of how to actually do them
• McCarthy (2007) is an extremely good, and extremely friendly, intro to Bayesian methods in ecology. It focuses on WinBUGS, and doesn't really cover hierarchical models in much detail, but I would recommend it as a first step.
• Clark (2007) is thorough, but a bit difficult to get into. I haven't looked at the accompanying lab manual, but I suspect that's probably one of the best places to start if you want to code your own MCMC routines in R (Clark recommends against WinBUGS, feeling that you can get yourself in too much trouble if you don't know what you're doing and that the best way to use these techniques is to code your own).
• Alberts Bayesian Computation in R (Springer, 2008) looks good too, but I (BMB) haven't looked at a copy (Ian Dworkin says "pretty useful book, and I would say makes a good companion to the Gelman et al (BDA) text. However [it] focuses less on writing your own gibbs sampler and understanding MHA, and uses his own R library (LearnBayes I believe).")
• Gelman and Hill (2006, CUP) is not as good if you want to learn to code your own, but is very good as a general introduction to hierarchical/multilevel modeling; Gelman et al Bayesian Data Analysis is the (one of the) bible(s) of Bayesian/MCMC analysis (Dworkin: BDA "has a lot more on MCMC and computation")

Bibliography

1. Albert, Jim. 2008. Bayesian Computation with R. 1st ed. Springer.

2. Clark, James S. 2007a. Models for Ecological Data: An Introduction. Princeton, NJ: Princeton University Press.

3. —-. 2007b. Statistical Computation for Environmental Sciences in R: Lab Manual for Models for Ecological Data. Princeton University Press.

4. Clark, James Samuel, and Alan E. Gelfand. 2006. Hierarchical Modelling for the Environmental Sciences: Statistical Methods. Oxford University Press.

5. Gelman, Andrew, John B. Carlin, and Hal S. Stern. 2003. Bayesian Data Analysis. CRC Press.

6. Gelman, Andrew, and Jennifer Hill. 2006. Data Analysis Using Regression and Multilevel/Hierarchical Models. Cambridge, England: Cambridge University Press, xx xx. http://www.stat.columbia.edu/~gelman/arm/.

7. McCarthy, M. 2007. Bayesian methods for ecology. Cambridge, England: Cambridge University Press.