Markov Chain Monte Carlo Methods: Theory and Applications - Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications - page 69

very least pseudo-converged. In most practical applications, this may be the best one
can hope for.

3.3.5.1 Time Series Plots

This is one of the simplest, yet also most effective convergence diagnostics as it
leverages the powerful pattern recognition capabilities built into the human brain. It
is performed by simply plotting a long time series of one or more parameters from
the multidimensional parameter set. A chain can be said to have converged when
it varies rapidly about a stable central value, not exhibiting a trend in the mean or
changes in spread. The drawback to this technique is that it is qualitative rather
than quantitative. An illustration of the sort of time series plots that are used in this
analysis can be see in Fig. 3.1 .

3.3.5.2 Running or Batch Moments

In addition to time series plots of the parameter values, convergence can also be
diagnosed from time series of the moments of the sample computed in batches as
the chain runs. This leverages the fact that the chain should eventually converge
to sampling the stable (invariant) target distribution and as such should produce
stable values of the posterior moments. Alternatively, comparison of moments for
randomly selected sets of sub-samples (batches) can be done and the result should
be similar to that of the running moments.

3.3.5.3 Multi-chain Convergence Diagnostics: The R-Statistic

Another method of assessing convergence leverages the information contained in
the differences between chains of a multi-chain MCMC simulation. This method is
based on a comparison of the variance (or other moments) within each chain to the
variance between chains for each estimated parameter, and is done in the following
manner ( Gelman et al. 2004 ). Consider

m chains, each of length n samples. First,

the within-chain variance is computed for each parameter x as

"

x ij x j 2 #

X n X

W D 1
m

1
n

; (3.10)

j D 1 i D 1

x j is the mean of each parameter x

where within each chain

n X

x j D 1
n

x ij : (3.11)

i D 1