Markov chain Monte Carlo for statistical inference

These notes provide an introduction to Markov chain Monte Carlo methods that are useful in both Bayesian and frequentist statistical inference. Such methods have revolutionized what can be achieved computationally, primarily but not only in the Bayesian paradigm. The account begins by describing ordinary Monte Carlo methods, which, in principle, have exactly the same goals as the Markov chain versions but can rarely be implemented. Subsequent sections describe basic Markov chain Monte Carlo, founded on the Hastings algorithm and including both the Metropolis method and the Gibbs sampler as special cases, and go on to discuss more recent developments. These include Markov chain Monte Carlo p–values, the Langevin–Hastings algorithm, auxiliary variables techniques, perfect Markov chain Monte Carlo via coupling from the past, and reversible jumps methods for target spaces of varying dimensions. Specimen applications, drawn from several different disciplines, are described throughout the notes. Several of these appear for the first time. All computations use APL as the programming language, though this is not necessarily a recommendation! The author welcomes comments and criticisms