Markov Chain Monte Carlo Methods for Statistical Inference

These notes provide an introduction to Markov chain Monte Carlo methods and their applications to both Bayesian and frequentist statistical inference. Such methods have revolutionized what can be achieved computationally, especially in the Bayesian paradigm. The account begins by discussing ordinary Monte Carlo methods: these have the same goals as the Markov chain versions but can only rarely be implemented. Subsequent sections describe basic Markov chain Monte Carlo, based on the Hastings algorithm and including both the Metropolis method and the Gibbs sampler as special cases, and go on to discuss some more specialized developments, including adaptive slice sampling, exact goodness–of–fit tests, maximum likelihood estimation, the Langevin–Hastings algorithm, auxiliary variables techniques, perfect sampling via coupling from the past, reversible jumps methods for target spaces of varying dimensions, and simulated annealing. Specimen applications are described throughout the notes.