Markov chain Monte Carlo

In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that is, the Markov chain's equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution.

Markov chain Monte Carlo methods are used to study probability distributions that are too complex or too highly dimensional to study with analytic techniques alone. Various algorithms exist for constructing such Markov chains, including the Metropolis–Hastings algorithm.

Markov chain Monte Carlo methods create samples from a continuous random variable, with probability density proportional to a known function. These samples can be used to evaluate an integral over that variable, as its expected value or variance.

Practically, an ensemble of chains is generally developed, starting from a set of points arbitrarily chosen and sufficiently distant from each other. These chains are stochastic processes of "walkers" which move around randomly according to an algorithm that looks for places with a reasonably high contribution to the integral to move into next, assigning them higher probabilities.

Random walk Monte Carlo methods are a kind of random simulation or Monte Carlo method. However, whereas the random samples of the integrand used in a conventional Monte Carlo integration are statistically independent, those used in MCMC are autocorrelated. Correlations of samples introduces the need to use the Markov chain central limit theorem when estimating the error of mean values.

These algorithms create Markov chains such that they have an equilibrium distribution which is proportional to the function given.

The development of MCMC methods is deeply rooted in the early exploration of Monte Carlo (MC) techniques in the mid-20th century, particularly in physics. These developments were marked by the Metropolis algorithm proposed by Nicholas Metropolis, Arianna W. Rosenbluth, Marshall Rosenbluth, Augusta H. Teller, and Edward Teller in 1953, which was designed to tackle high-dimensional integration problems using early computers. Then in 1970, W. K. Hastings generalized this algorithm and inadvertently introduced the component-wise updating idea, later known as Gibbs sampling. Simultaneously, the theoretical foundations for Gibbs sampling were being developed, such as the Hammersley–Clifford theorem from Julian Besag's 1974 paper. Although the seeds of MCMC were sown earlier, including the formal naming of Gibbs sampling in image processing by Stuart Geman and Donald Geman (1984) and the data augmentation method by Martin A. Tanner and Wing Hung Wong (1987), its "revolution" in mainstream statistics largely followed demonstrations of the universality and ease of implementation of sampling methods (especially Gibbs sampling) for complex statistical (particularly Bayesian) problems, spurred by increasing computational power and software like BUGS. This transformation was accompanied by significant theoretical advancements, such as Luke Tierney's (1994) rigorous treatment of MCMC convergence, and Jun S. Liu, Wong, and Augustine Kong's (1994, 1995) analysis of Gibbs sampler structure. Subsequent developments further expanded the MCMC toolkit, including particle filters (Sequential Monte Carlo) for sequential problems, Perfect sampling aiming for exact simulation (Jim Propp and David B. Wilson, 1996), RJMCMC (Peter J. Green, 1995) for handling variable-dimension models, and deeper investigations into convergence diagnostics and the central limit theorem. Overall, the evolution of MCMC represents a paradigm shift in statistical computation, enabling the analysis of numerous previously intractable complex models and continually expanding the scope and impact of statistics.

Suppose (X_n) is a Markov Chain in the general state space ${\displaystyle {\mathcal {X}}}$ with specific properties. We are interested in the limiting behavior of the partial sums: