WE HAVE SEEN in the previous chapter that there exist methods to generate independent observations from the standard probability distributions, including those described in the appendix. But we still have the problem of what to do when faced with a nonstandard distribution such as the posterior distribution of parameters of the conditionally conjugate linear regression model. Although the methods described before can, in principle, deal with nonstandard distributions, doing so presents major practical difficulties. In particular, they are not easy to implement in the multivariate case, and finding a suitable importance function for the importance sampling algorithm or a majorizing density for the AR algorithm may require a very large investment of time whenever a new nonstandard distribution is encountered.

These considerations impeded the progress of Bayesian statistics until the development of Markov chain Monte Carlo (MCMC) simulation, a method that became known and available to statisticians in the early 1990s. MCMC methods have proved to be extremely effective and have greatly increased the scope of Bayesian methods. Although a disadvantage of this family of methods is that it does not provide independent samples, it has the great advantage of flexibility: it can be implemented for a great variety of distributions without having to undertake an intensive analysis of the special features of the distribution. We note, however, that an analysis of the distribution may shed light on the best algorithm to use when more than one is available.