Markov Chain Monte Carlo Methods: Theory and Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 57

Chapter 3
Markov Chain Monte Carlo Methods: Theory
and Applications

Derek J. Posselt

Abstract Markov chain Monte Carlo algorithms constitute flexible and powerful
solutions to Bayesian inverse problems. They return a sample of the unapproximated
posterior probability density, and make no assumptions as to linearity or the form
of the prior or likelihood. MCMC algorithms are in principle easy to construct,
however, they can prove difficult to implement in practice. This chapter describes
the theory that underlies MCMC simulation, provides guidance for its practical
implementation, and presents examples of applications of MCMC to satellite
retrievals and model uncertainty characterization. Though the high dimensionality
of Earth system datasets and the complexity of atmospheric, oceanic, and hydrologic
models present significant challenges, continued advances in theory and practice are
making application of MCMC algorithms increasingly feasible.

3.1 Introduction and History

Markov chain Monte Carlo (MCMC) algorithms were born out of investigations
into numerical integration at Los Alamos national laboratory in the 1940s and
1950s. These were initially centered around the development of Monte Carlo (MC)
methods, which comprise a class of algorithms designed to compute numerical
solutions to integrals using random draws from a specified probability distribution.
MC algorithms were developed as a way to numerically solve neutron diffusion
problems during the development of the atomic bomb, but were generally limited to
low-dimensional problems. Shortly after the advent of Monte Carlo based random
simulation, Metropolis et al. ( 1953 ) developed an extension that allowed MC to be
used to evaluate integrals over large dimensional spaces. The method was used to