Markov Chain Monte Carlo Methods: Theory and Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 58

simulate the movement of interacting particles in equilibrium with each other. In
essence, the Metropolis algorithm is composed of a random sequence of particle
moves, each of which depends only on the current position (thus forming a Markov
chain). In each new configuration of particles, the energy of the system is computed
and compared with the energy of the previous configuration. If the new state has
lower energy than the old, it is accepted as a valid particle configuration. If not, it
is accepted with a probability that depends on the difference in energy between the
two states. The collection of states that results from a sufficiently long sequence of
proposed configurations describes a system of particles in equilibrium with correct
relationship between pressure, temperature, and volume. Hastings ( 1970 ) later
generalized the Metropolis algorithm for use in integrating probability distributions,
and this constituted the first use of Markov chain Monte Carlo algorithms for
posterior probability simulation.
It is thought that the relative lack of computational power limited the general
applicability of MCMC for statistical computation until the 1980s. Geman and
Geman ( 1984 ) applied a variant of MCMC to the problem of image restoration and
drew an analogy between image processing and computation of posterior probability
densities. The use of Gibbs random fields in this study resulted in their algorithm
being termed the “Gibbs sampler”. Inspired by the work of Geman and Geman
( 1984 )and Gelfand and Smith ( 1990 ) generalized MCMC-based computation of
posterior probability densities from of a collection of algorithms that included
Geman and Geman ( 1984 )'s Gibbs sampler, data augmentation methods ( Tanner and
Wong 1987 ), and importance sampling ( Rubin 1987 ). It is generally acknowledged
that the Gelfand and Smith ( 1990 ) paper led to widespread use of MCMC for
Bayesian posterior sampling in the statistical community, and that Tierney ( 1994 )
was the first to thoroughly describe the necessary convergence properties of the
underlying Markov chains. Building on the foundations laid by Metropolis et al.
( 1953 ), Hastings ( 1970 ), Gelfand and Smith ( 1990 ), Tierney ( 1994 ), and Green
( 1995 ) further generalized the MCMC algorithm to the exploration of probability
spaces with variable dimensions. Since the mid-1990s, a host of variants of MCMC
have been proposed, most aimed at increasing the efficiency with which MCMC
samples the posterior probability space.
Though the use of MCMC in statistical computation is now common ( Gelman
et al. 2011 ), its adoption in the atmospheric and oceanic sciences has been slow.
This is in part due to the exceedingly large dimensionality of most atmospheric and
oceanic state estimation problems, as well as to the complexity and computational
expense of geophysical process (forward) models. In the remainder of this chapter,
we will briefly present the theory that underlies MCMC algorithms, describe the
practical issues users encounter when implementing MCMC for a new application,
highlight the successful use of MCMC in satellite retrievals and model parameter
estimation, and finish with concluding remarks as to future applications of MCMC
in the atmospheric sciences.