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Monte Carlo Method

Markov chain Monte Carlo is an important technique used by Bayesian practitioners to sample from the posterior distribution. The Monte Carlo method is, in general terms, any technique used for obtaining solutions to deterministic problems using random numbers. The term Monte Carlo was coined by von Neumann and Ulam in the 1940's in the context of such problems [34].

A simple example of this [25] is the evaluation of the following integral;

displaymath2283

Analytical solution of the above is difficult, but Monte Carlo simulation proposes the following;

  1. Let i = 0; Let N be some large number.
  2. Sample tex2html_wrap_inline2291 from the exponential so tex2html_wrap_inline2293
  3. Let tex2html_wrap_inline2295 if tex2html_wrap_inline2297 and 0 otherwise
  4. Let i=i+1. If i<N return to step 2.
  5. Then I is estimated by tex2html_wrap_inline2305 .

Observe that the above is the standard estimator for tex2html_wrap_inline2307 . In practice, many of the values of interest are expected values. To obtain posterior expectations of a function of our parameter, tex2html_wrap_inline2309 , we need to calculate integrals of the type

displaymath2284

It is possible to use the above idea of Monte Carlo methods, importance sampling, together with some Markov Chain theory, to efficiently approximate such expressions. Some theory is outlined below.


Cathal Walsh
Sat Jan 22 17:09:53 GMT 2000