Bayesian computation (MCMC)

Abstract This chapter provides a detailed introduction to modern Bayesian computation. The Metropolis–Hastings algorithm is illustrated using a simple example of distance estimation between two sequences. A number of generic Markov chain Monte Carlo (MCMC) proposal moves are described, and the calculation of their proposal ratios is illustrated. The chapter discusses the convergence rate of the Markov chain as well as its mixing efficiency, as influenced by the MCMC proposal. The chapter also illustrates several advanced MCMC algorithms, including parallel tempering (Metropolis-coupled MCMC or MCMCMC) which uses heated chains to improve mixing when there are multiple local peaks on the posterior surface, reversible jump MCMC (rjMCMC) which is used in trans-model and trans-dimensional inference, and calculation of the Bayes factor used in Bayesian model selection.