Statistical Methods in Medical Research 4ed

Markov chain Monte Carlo methods, orMCMC methods, has proved to be of
great value.
The `Monte Carlo' part of MCMC arises because the method is essentially
based on the generation of random samplesÐa similar use to that encountered in
the context of permutation tests (see §10.6). The `Markov chain' element of
MCMC refers to the theoretical underpinning of the method and this will be
explained very briefly later. However, before this, the Gibbs sampler, perhaps the
most intuitively accessible MCMC method, will be described.
MCMC methods are currently the subject of intensive research and all
aspects of the subject, theoretical, practical and computational, are developing
very rapidly. The present description is at a very naõÈve level and the reader
interested in deeper explanations should consult the current literature: Besag et
al. (1995) Gilks et al. (1996) and Brooks (1998) are good starting-points in a
literature that is already large and likely to expand much further in the next few
years.
The Gibbs sampler
16.4 Markov chain Monte Carlo methods 549
The Gibbs sampler is perhaps the most widely used MCMC method. In the form
most commonly used today it was introduced by Geman and Geman (1984) in a
specialized context to do with image analysis using distributions known as Gibbs
distributions. This explains what is an unfortunate aspect of the nomenclature,
because the method can be applied to a much wider range of distributions. It is
also a special case of methods that were introduced much earlier by Metropolis et
al. (1953) and Hastings (1970), which will be explained later.
The problem is how to draw samples from a distribution, p…ujy† whose
argument is a k-dimensional vector, and which will often be of a non-standard
form. The Gibbs sampler uses the k so-called full conditional distributions, which
are the conditional distributions
p…u1ju 1, y†
p…u2ju 2, y†
.
p…ukju k, y†:
The distributions are full conditionals because all the parameters other than the
argument are included in the conditioning event. If the problem were specified in
terms of three parameters, u1, u2, u3 then p…u1ju2, u3, y† is a full conditional
whereas p…u1ju2, y† is not.
The analyst must start the Gibbs sampler by providing an arbitrary starting
vector u …0† . The sampler then proceeds to generate a sequence of vectors
u …1† , u …2† , u …3† , ..., u …N† where N is under the control of the analyst (although in