Statistical Methods in Medical Research 4ed

close to the stationary distribution. The length, M, of this burn-in period is
chosen in the belief that the distribution of the later draws from the chain will be
close to the stationary distribution and we can consider that, for all practical
purposes, the chain has converged by time M. The length of the chain that is used
to explore the posterior distribution, N M, should be controlled by choosing N
so that the precision of the estimate obtained from (16.15) is adequate.
There are practical difficulties in assessing the adequacy of (16.15), because of
the dependence between the u …i† , and time series methods (see §12.7) are often
used (see Geyer, 1992). Even once a Markov chain has, for all practical purposes,
converged, it may be slow to mix. This means that the chain evolves slowly and
takes many iterations to move round the state space. This can occur when the
correlation between successive outputs from the chain are highly correlated.
Unless such chains are run for very long periods, they will give results that
have poor precision. The parameterization chosen for the model can lead to
problems with mixing, and reparameterization can be a useful remedy. This and
other ways to improve the practical performance of MCMC methods are discussed
in Gilks and Roberts (1996).
Metropolis±Hastings algorithms
16.4 Markov chain Monte Carlo methods 557
Starting from arbitrary initial values, the Gibbs sampler indicates how to simulate
the next draw from the posterior, u …i‡1† , from the present draw u …i† . In the
simple form described above, each element of u …i‡1† is updated by taking a
sample from the appropriate full conditional distribution. The Gibbs sampler
is, in fact, a special case of a more general algorithm for producing MCMC
samples. This algorithm was originally due to Metropolis et al. (1953) and
extended by Hastings (1970)Ðit is now generally referred to as the MetropolisÐHastings
algorithm.
The algorithm has aspects which are reminiscent of the method of rejection
sampling discussed in §16.3, in that the generation of u …i‡1† requires the specification
of a proposal distribution from which to draw a candidate value, uC. The
algorithm ensures the correct distribution for u …i‡1† by accepting uC (i.e. setting
u …i‡1† ˆ uC) with an astutely chosen probability. However, unlike the rejection
sampling method, multiple draws are not needed in order to determine u …i‡1† Ðif
uC is not accepted, then u …i‡1† ˆ u …i† .
The results will be valid for any proposal distribution, although the closer the
proposal distribution approximates the posterior, the more efficient will be the
method. The posterior is p…ujy† and the proposal distribution must have a
density which is a function of a vector of the same dimension as u, q…uju …i† †,
which can depend on both y and u …i† , although dependence on y has been
suppressed in the notation. The candidate value, uC, is drawn from this distribution
and is accepted as u …i‡1† with probability