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the possibility arises that one or more of the chains may not have converged to
sampling the target distribution, and the resulting sample will not be representative
of the true posterior PDF. This is related to the problem of “pseudo-convergence”,
which is discussed in greater detail below (and it should also be noted that multiple
chains can be effectively used to diagnose convergence). Another common practice
is to start multiple chains in as widely dispersed locations in the parameter space
as possible, creating an “over-dispersed” initial sample (
Gelman et al. 2004
). As
discussed above, it is not desirable to start a chain in a region of the space that
contains low probability values as it may then take more time for the algorithm
to find a region with relatively large probability density, and in the process the
proposal may be badly mis-tuned. In most computing environments, and for most
applications, it is not possible to disperse the initial points in such a way as they
span the full range of possible parameter combinations. As such, some (potentially
vast) regions of the space will not contain a Markov chain start point and may be
left unexplored.
3.3.4
Pseudo-convergence
Because the true structure of the posterior PDF is not typically known in advance,
the possibility exists that a chain that has appeared to have converged may in fact
have simply spent a very long time sampling a localized probability structure. After
running for a suitably long interval, the chain may make a sudden transition to a
new and previously unexplored high probability region of the space. This situation
is most common for posterior PDFs that exhibit multiple highly localized modes
and is referred to as pseudo-convergence; the tendency for the chain in this case to
appear to have converged to sampling the invariant target distribution when in reality
it has not. In practice, there is no way to guarantee the chain has truly converged;
the best way to safeguard against pseudo-convergence is to run very long chains.
That said, it is possible that the use of a heavy-tailed proposal distribution may
help to avoid this problem by making large proposed moves with greater frequency
than the centered multivariate Normal. It may also be possible to make clever use
of proposals from multiple chains to increase the likelihood of jumping between
widely dispersed modes (e.g.,
Vrugt et al. 2009
;
Vrugt and Ter Braak 2011
).
3.3.5
Diagnosing Convergence
As mentioned above, in cases for which the shape of the true posterior PDF is
unknown, it is impossible to know with absolute certainty that the Markov chain
has converged to sampling the invariant target distribution. Even so, there are several
diagnostic tools that can be brought to bear in assessing whether the chain has at the
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