Markov chain Monte Carlo

In statistics, Markov Chain Monte Carlo (MCMC) methods are a class of algorithms for sampling from a probability distribution based on constructing a Markov chain that has the desired distribution as its equilibrium distribution. The state of the chain after a 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.

Convergence of the Metropolis-Hastings algorithm. MCMC attempts to approximate the blue distribution with the orange distribution

Random walk Monte Carlo methods make up a large subclass of MCMC methods.

Application domains

Classification

Random walk Monte Carlo methods

See also: Random walk

Multi-dimensional integrals

When an MCMC method is used for approximating a multi-dimensional integral, an ensemble of "walkers" move around randomly. At each point where a 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 a reasonably high contribution to the integral to move into next.

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 methods are correlated. A Markov chain is constructed in such a way as to have the integrand as its equilibrium distribution.

Examples

Examples of random walk Monte Carlo methods include the following:

Other MCMC methods

Markov Chain quasi-Monte Carlo (MCQMC)[5][6] The advantage of low-discrepancy sequences in lieu of random numbers for simple independent Monte Carlo sampling is well-known.[7] This procedure, known as Quasi-Monte Carlo method (QMC),[8] yields an integration error that decays at a superior rate to that obtained by IID sampling, by the Koksma-Hlawka inequality. Empirically it allows to reduce both estimation error and convergence time by an order of magnitude.

Reducing correlation

More sophisticated methods use various ways of reducing the correlation between successive samples. These algorithms may be harder to implement, but they usually exhibit faster convergence (i.e. fewer steps for an accurate result).

Examples

Examples of non-random walk MCMC methods include the following:

Convergence

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.

Typically, 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.

Many random walk 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.

See also

Notes

  1. ↑ See Gill 2008.
  2. ↑ See Robert & Casella 2004.
  3. ↑ Banerjee, Sudipto; Carlin, Bradley P.; Gelfand, Alan P. Hierarchical Modeling and Analysis for Spatial Data (Second Edition ed.). CRC Press. p. xix. ISBN 978-1-4398-1917-3.
  4. ↑ See Green 1995.
  5. ↑ Chen, S., Josef Dick, and Art B. Owen. "Consistency of Markov chain quasi-Monte Carlo on continuous state spaces." The Annals of Statistics 39.2 (2011): 673-701.
  6. ↑ Tribble, Seth D. Markov chain Monte Carlo algorithms using completely uniformly distributed driving sequences. Diss. Stanford University, 2007.
  7. ↑ Papageorgiou, Anargyros, and J. F. Traub. "Beating Monte Carlo." Risk 9.6 (1996): 63-65.
  8. ↑ Sobol, Ilya M. "On quasi-monte carlo integrations." Mathematics and Computers in Simulation 47.2 (1998): 103-112.
  9. ↑ See Neal 2003.
  10. ↑ See Stramer 1999.

References

Further reading

External links