Statistical Methods in Medical Research 4ed

556 Further Bayesian methods
recurrence is a property that ensures that the chain can keep getting to all parts
of the space. Aperiodicity stops the chain from cycling within a subset of the
state space. See Roberts and Smith (1994) and Roberts (1996) for more precise
and thorough discussions.
In an application of MCMC methods, the elements of the chain, Xi, will
usually correspond to the parameters in the modelÐin other words, the Xi will be
vectors. Moreover, most of the parameters will be real numbers so the state space
of the Markov chain will not be the simple finite set discussed so far, but a
suitable subset of k-dimensional space. At the very informal level of the foregoing
discussion, there is little difference between Markov chains with discrete or
continuous state spaces. At deeper levels, the technicalities of general state spaces
are more demanding, and some of the concepts need to be amended; see Tierney
(1996) for an illuminating introduction.
From a practical point of view, MCMC is useful only if the output from the
Markov chain can be used to estimate useful summaries of the posterior distribution.
Many estimators can be written in the form
1
N
P N
`ˆ1
f …u …`† †, …16:14†
where u …1† , u …2† , u …3† , ..., u …N† denotes the output from an MCMC procedure and
f … † is an appropriately chosen function. For example, if the intention were to
estimate the marginal posterior mean of the first element of u, then f would be
defined by f …u† ˆu1. Iff …u† ˆ1ifu2 < c and 0 otherwise, then (16.14) would
be used to estimate the posterior probability that u2 < c.
While it is well known that means of the form (16.14) will be good estimators
of the posterior expectation of f …u† in straightforward circumstancesÐfor example,
when the u …1† , u …2† , u …3† , ..., u …N† are independent and come from the same
distributionÐthis will not be the case for MCMC output. There will be
dependence between the different elements of the sequence and the elements
will not share a common distribution. However, as N gets larger, the later
elements of the sequence will all come close to having the stationary distribution
and, if the stationary distribution is the required posterior distribution, it can be
shown that (16.14) does provide a valid estimate of the posterior expectation of
f …u†.
Many MCMC practitioners would replace (16.14) by
1
N M
P N
`ˆM‡1
f …u …`† †: …16:15†
They would omit the information from the first M draws from the Markov chain
on the grounds that at those times the chain had not been running for sufficiently
long for it to be reasonable to assume that the draws would have distributions