Statistical Methods in Medical Research 4ed

An important idea, although not one that is universally accepted, is to run
more than one chain. The advantages of many chains over one much longer
chain have been discussed widely (see Gelman & Rubin, 1992, and associated
following discussion). In any given application the issue may be resolved by
practical considerations of the computational equipment available. However,
multiple chains clearly allow the possibility of starting chains from a variety of
values and assessing how long the influence of the starting value appears to
persist. Gelman (1996) argues that methods which try to assess convergence on
the basis of a single sequence from an MCMC method often fail to detect nonconvergence.
Inference
16.4 Markov chain Monte Carlo methods 559
Once the length of the `burn-in' period has been determined, it remains to decide
how many further samples should be used to make inferences about the posterior
distribution. Essentially this is just a problem in estimation for which standard
sample-size calculations might be contemplated. If the aim is to estimate the
posterior mean of some scalar function, f …u†, of the parameter, then generating
enough values from the posterior to ensure that the standard error of this
quantity is below a specified value might be a sensible approach. If an estimate
of the standard deviation, s, off…u† is available, then, assuming that the values
obtained from the chain are independent, the usual formula for the standard
error, s= n
p can be used to estimate the number, n, of simulations required.
However, it will often be inappropriate to assume that the values obtained
from the chain are independent and time series methods can be applied. For
example, a first-order autoregression (see §12.7) can be fitted to the successive
values of f …u† obtained from the chain. If this approach is followed then the
appropriate formula for the standard error becomes
r
s
p
n
1 ‡ r
1 r
where r is the autocorrelation coefficient. It can be appreciated from this
formula that, if there is substantial autocorrelation in the series, so that r is
close to 1, a very large value of n may be needed to ensure that the standard error
is below the required value. A valuable discussion of related issues can be found
in Geyer (1992). See also Raftery and Lewis (1992).
If it is necessary to use a very long chain to obtain adequate estimates of
relevant quantities, then problems can arise with the amount of computer
storage required. If the successive values are highly correlated (and, as just
seen, very long runs are likely to be required in just this circumstance), perhaps
little will be lost by keeping only every Lth value of u obtained from the chain.
Applying this device, which is known as thinning, may well result in a set of
,