M arkov Chain M onte C arlo, Introduction

Abstract

Abstract In this article, we introduce the basic ideas of Markov chain Monte Carlo simulation. We start by briefly commenting on the main ideas of Monte Carlo sampling and then by reviewing the theory of Markov chains, concentrating in particular on the conditions required for the existence of a stationary distribution. We then introduce the Metropolis–Hastings algorithm and show that this generates a Markov chain with a specific stationary distribution. We next introduce some specific ways of implementing the Metropolis–Hastings algorithm, via the independence sampler, the Metropolis sampler and the Gibbs sampler among others. Finally, we comment on how the convergence of a Markov chain to equilibrium can be assessed in practice and we finish with an example.