Metropolis Algorithm

Definition

The Metropolis algorithm is a Monte Carlo method advanced by Metropolis et al. (1953) to generate samples from a prespecified target probability distribution. Originally, it was applied to investigate the statistical mechanics of fluids. By now, this method and its extensions are used for a wide range of problems in scientific computing (see, e.g., Liu (2004)). The basic idea is to simulate a Markov chain so that its stationary distribution is the target distribution.

Let ${\mathbb {X}}$ be a discrete state space (finite or countable) on which the target probability distribution $\pi = {\left( {{\pi_x}}\right)_{{x \in {\mathbb {X}}}}}$ is defined. It is assumed that ${\pi_x}> 0,\;x \in {\mathbb {X}}$. Suppose that Q is a symmetric transition probability matrix, that is $Q = {\left( {{q_{{xy}}}} \right)_{{x,y \in {\mathbb {X}}}}}$ with q _xy  ≥ 0, q _xy  = q _yx , $\sum\nolimits_{{y \in {\mathbb {X}}}} {{q_{{xy}}}} = 1,x,y \in {\mathbb {X}}$. The following...