The Metropolis Algorithm

The author reviews the Metropolis et al. (1953) algorithm to generate any given probability distribution. First, the author discusses the relevant aspects of the theory of Markov processes and proves convergence theorems that guarantee that the algorithm converges to a unique ensemble. Next, some applications that illustrate the use of the algorithm are discussed. Appendix 1 contains the statement and proof of a theorem due to Perron (1907) on matrices with positive elements which is important in the proof of convergence. Appendix 2 contains the example of the generation of a one-dimensional distribution using the Metropolis and heat-bath algorithms.