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Bayesian Psychometric Modeling (2016) by Roy Levy and Robert J. Mislevy

5 Markov Chain Monte Carlo Estimation

This chapter on MCMC methods gives an introduction to some of the common and basic sampling approaches for Bayesian methods. These methods in

  1. Gibbs Sampling

  2. Metropolis Sampling

  3. Metropolis-Hastings

and some notes on how these approaches are related. The most important take away for me was their section on practical issues in MCMC methods. These practical aspects of estimation that should be noted are:

  1. Assessing convergence - making sure enough iterations have been used including the potential scale reduction factor (\(\hat{R}\)),

  2. Serial dependence - where the samples drawn from the posterior are autocorrelated. This means that within a chain the draws are dependent but with enough draws and thinning all samples are sufficiently independent,

  3. Mixing - that different chains search/sample from the same parameter space but different chains can sometimes get “stuck” sampling one part of the parameter space that is not the same as the other chains.

Lastly, a major take away from this chapter is that MCMC methods help to approximate the posterior distribution. The distribution is the solution of a full Bayesian analysis and not a point estimate.