The theory of the Metropolis-Hastings algorithm

Abstract

Markov Chain Monte Carlo is a general method used when it is required to sample from some unknown distribution which is complex, non-standard and known only up to a normalizing constant. With the help of these techniques, a Markov Chain is formed, whose distribution at stationarity is the same as the target distribution to be sampled from. The Metropolis-Hastings algorithm is the most general of these algorithms and provides a method for selecting the candidate draws which are to form the Markov Chain. These are determined by an acceptance probability, where if the current draw is rejected, the following step remains equivalent to the current state of the chain. The effectiveness of this algorithm and its speed of convergence depend on many factors, including the number of iterations the algorithm is to be run for, the choice of the starting distribution and how well it approximates the target distribution and the starting point. Multiple convergence diagnostics exist for these methods. These can be used either to determine whether the Markov Chain has reached equilibrium or the number of iterations necessary to ensure that the results obtained from the chain are of the required precision. However, no one such diagnostic exists which can guarantee that the chain has reached convergence and thus a combination of methods must be used. The objective of this thesis is to discuss the ideal choice of parameters for the algorithm as well as to analyse some of the most popular diagnostics.