ABSTRACT
This chapter describes certain specific algorithms that have become the dominant approaches in Bayesian psychometric modeling with Markov chain Monte Carlo (MCMC). It provides an overview of MCMC estimation, and broad description and comparison of MCMC to frequentist estimation. The chapter illuminates the conceptual alignment between MCMC and Bayesian statistical modelling. In a Bayesian analysis, the posterior distribution is the solution obtained from fitting the model to the data. In complex Bayesian models, the posterior distribution is often difficult to obtain through analytical methods, and not often easier to sample from directly. Inspection of the Metropolis sampler reveals that the posterior distribution appears in both the numerator and denominator of the acceptance probability and therefore only needs to be known up to a constant of proportionality.
