Markov chain Monte Carlo

Markov chain Monte Carlo (MCMC) methods (which include random walk Monte Carlo methods) are a class of algorithms for sampling from probability distributions based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a large number of steps is then used as a sample of the desired distribution. The quality of the sample improves as a function of the number of steps.

Usually it is not hard to construct a Markov chain with the desired properties. The more difficult problem is to determine how many steps are needed to converge to the stationary distribution within an acceptable error. A good chain will have rapid mixing—the stationary distribution is reached quickly starting from an arbitrary position—described further under Markov chain mixing time.

Typical use of MCMC sampling can only approximate the target distribution, as there is always some residual effect of the starting position. More sophisticated MCMC-based algorithms such as coupling from the past can produce exact samples, at the cost of additional computation and an unbounded (though finite in expectation) running time.

The most common application of these algorithms is numerically calculating multi-dimensional integrals. In these methods, an ensemble of "walkers" moves around randomly. At each point where the walker steps, the integrand value at that point is counted towards the integral. The walker then may make a number of tentative steps around the area, looking for a place with reasonably high contribution to the integral to move into next. Random walk 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 correlated. A Markov chain is constructed in such a way as to have the integrand as its equilibrium distribution. This is often easy to do.

Multi-dimensional integrals often arise in Bayesian statistics, computational physics, computational biology and computational linguistics, so Markov chain Monte Carlo methods are widely used in those fields. For examples, see Gill and Robert & Casella.

Random walk algorithms
Many Markov chain Monte Carlo methods move around the equilibrium distribution in relatively small steps, with no tendency for the steps to proceed in the same direction. These methods are easy to implement and analyze, but unfortunately it can take a long time for the walker to explore all of the space. The walker will often double back and cover ground already covered. Here are some random walk MCMC methods:


 * Metropolis–Hastings algorithm: Generates a random walk using a proposal density and a method for rejecting proposed moves.
 * Gibbs sampling: Requires that all the conditional distributions of the target distribution can be sampled exactly. Popular partly because when this is so, the method does not require any 'tuning'.
 * Slice sampling: Depends on the principle that one can sample from a distribution by sampling uniformly from the region under the plot of its density function. This method alternates uniform sampling in the vertical direction with uniform sampling from the horizontal 'slice' defined by the current vertical position.
 * Multiple-try Metropolis: A variation of the Metropolis–Hastings algorithm that allows multiple trials at each point. This allows the algorithm to generally take larger steps at each iteration, which helps combat problems intrinsic to large dimensional problems.

Avoiding random walks
More sophisticated algorithms use some method of preventing the walker from doubling back. These algorithms may be harder to implement, but may exhibit faster convergence (i.e. fewer steps for an accurate result).


 * Successive over-relaxation: A Monte Carlo version of this technique can be seen as a variation on Gibbs sampling; it sometimes avoids random walks.
 * Hybrid Monte Carlo (HMC): Tries to avoid random walk behaviour by introducing an auxiliary momentum vector and implementing Hamiltonian dynamics where the potential energy function is the target density. The momentum samples are discarded after sampling.  The end result of Hybrid MCMC is that proposals move across the sample space in larger steps and are therefore less correlated and converge to the target distribution more rapidly.
 * Some variations on slice sampling also avoid random walks.
 * Langevin MCMC and other methods that rely on the gradient (and possibly second derivative) of the log posterior avoid random walks by making proposals that are more likely to be in the direction of higher probability density.

Changing dimension
The reversible-jump method is a variant of Metropolis–Hastings that allows proposals that change the dimensionality of the space. This method was proposed in 1995 by Peter Green of Bristol University. Markov chain Monte Carlo methods that change dimensionality have also long been used in statistical physics applications, where for some problems a distribution that is a grand canonical ensemble is used (e.g., when the number of molecules in a box is variable). Some sort of reversible-jump variant is also needed when doing MCMC or Gibbs sampling over nonparametric Bayesian models such as those involving the Dirichlet process or Chinese restaurant process, where the number of mixing components/clusters/etc. is automatically inferred from the data.