Markov Chain Monte Carlo (MCMC)

The term Monte Carlo method (suggested by John von Neumann and S. M. Ulam, in the 1940's) refers to simulation of processes, using random numbers. The term Monte Carlo (a city long known for its gambling casinos) derived from the fact that "numbers of chance" (i.e., Monte Carlo simulation methods) were used in order to solve some of the integrals of the complex equations involved in the design of the first nuclear bombs (integrals of quantum dynamics). By generating large samples of random numbers from, for example, mixtures of distributions, the integrals of these (complex) distributions can be approximated from the (generated) data.

Complex equations with difficult-to-solve integrals are often involved in Bayesian Statistics Analyses. For a simple example of the MCMC method for generating bivariate normal random variables, see the description of the Gibbs Sampler.

STATISTICA Visual Basic provides an excellent environment for performing these types of analyses, because it 1) includes an efficient and high quality random number generator, 2) can quickly generate results spreadsheets and various graphs from the simulated data, and 3) allows convenient access to external (compiled) routines for very fast data processing.

For a detailed discussion of MCMC methods, see Gilks, Richardson, and Spiegelhalter (1996). See also the description of the Gibbs Sampler, and Bayesian Statistics (Analysis).