Monte Carlo Methods and Bayesian Computation: MCMC

Markov chain Monte Carlo (MCMC) methods use computer simulation of Markov chains in the parameter space. The Markov chains are defined in such a way that the posterior distribution in the given statistical inference problem is the asymptotic distribution. This allows to use ergodic averages to approximate the desired posterior expectations. Several standard approaches to define such Markov chains exist, including Gibbs sampling, Metropolis-Hastings and reversible jump. Using these algorithms it is possible to implement posterior simulation in essentially any problem which allow pointwise evaluation of the prior distribution and likelihood function.