Markov Chain Monte Carlo Signal Processing
Markov chain Monte Carlo refers to a class of algorithms that
allow one to draw (pseudo-) random samples from an arbitrary target probability
density, p(x), known up to a normalizing constant. The basic
idea behind these algorithms is that one can achieve the sampling from the
target density p(x) by running a Markov chain whose equilibrium
density is exactly p(x). Here we describe two basic MCMC algorithms,
the Metropolis algorithm and the Gibbs sampler, which have been widely used in
diverse fields. The validity of both algorithms can be proved using basic Markov
chain theory [420].
The roots of MCMC methods can be traced back to the well-known
Metropolis algorithm [318], which was used initially to
investigate the equilibrium properties of molecules in a gas. The first use of
the Metropolis algorithm in a statistical context is found in [174]. The Gibbs sampler, which is a
special case of the Metropolis algorithm, was so termed in the seminal paper [137] on image
processing. It was brought to statistical prominence by [134], where it was observed that many
Bayesian computations could be carried out via the Gibbs sampler. For tutorials
on the Gibbs sampler, see [21, 68].
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