Markov Chain Monte Carlo Methods

Markov chain Monte Carlo (MCMC) methods use a carefully constructed sequence of dependent random variables to approximate expectations with respect to probability distributions of interest (i.e., appropriately weighted averages). Although first developed in the 1940s by physicists at Los Alamos, NM, the interest in these methods within the field of educational research is a consequence of their broad applicability to statistical inference. In many modern statistical problems, it is necessary to either find the maximum of a complicated function (as in maximum likelihood estimation) or compute high dimensional integrals (as in many forms of Bayesian inference). Monte Carlo methods provide approximation schemes suitable for addressing both of these problems when analytical or simpler numerical schemes fail; MCMC methods are perhaps the most broadly used class ...