ABSTRACT
In this chapter, the authors introduce Markov chain Monte Carlo (MCMC), a powerful collection of algorithms that enable to simulate from complicated distributions using Markov chains. The development of MCMC has revolutionized statistics and scientific computation by vastly expanding the range of possible distributions that they can simulate from, including joint distributions in high dimensions. The Metropolis-Hastings algorithm is an extremely general way to construct a Markov chain with a desired stationary distribution. The MCMC approach is to obtain a large number of draws from a Markov chain whose stationary distribution is the posterior distribution. The Metropolis-Hastings algorithm uses any irreducible Markov chain on the state space to generate proposals, then accepts or rejects those proposals so as to produce a modified Markov chain with the desired stationary distribution. The choice of the proposal distribution is extremely important in practice, since a bad proposal distribution may result in very slow convergence to the stationary distribution.
